Her baptism:

[                       Glasgow Parish    Ref 644/1

“                          Nov 9 1776                     271
“. . . .
“Brown     Bailey John Brown and jean Mayn a SD Frances born Oct 8  Wit. Alexr Monro and Tho Napier
___________________________


From the following she had the name of Frances Allen Brown.  I wonder where the "Allen" came from?  Her brother Allan?  But Allen was not in her marriage index records nor the index records for the baptisms of any of her 10 children.

Found at:
  http://www.electricscotland.com/history/canada/campbell_archibald.htm
  _________________________________________________________
  

                           Canadian History
                      
                      Archibald Hamilton Campbell

Of Toronto and born on 12th August, 1819, at Carbrook, Stirlingshire, Scotland, the country seat of his father, John Campbell, W.S., who was born in 1770, and who was the great grandson and representative in the male line of John Campbell of Easter Shian and Garrows, in the County of Perth, grandson of John Campbell of Edramuckie Castle, on Loch Tay. Mr. Campbell of Carbrook married, in 1794, Francis Allen Brown, grand-daughter of William Mayne, of Powis and Logie, Counties of Perth and Clackmanan, the great-grandson of William Mayne of Pile, who lived in the time of Mary and James VI., was succeeded by his son John, born 1586, and who died in 1696, at the age of 110. (Vide Burke's Extinct Peerages, art. Mayne.) Our subject was educated at the Edinburgh Academy and the University of Edinburgh, at both of which the studies were chiefly classical. He always was fond of out-door sports and athletic exercises, and in studies of natural philosophy and mechanics. On leaving the University he adopted banking as a profession, but subsequently took up the study of Scotch law. In 1845 he came to Canada, and on the copper mining excitement breaking out, became connected with a mining enterprise on Lake Huron, which, like other similar ventures, was unsuccessful. Being offered the post of manager in a bank in Montreal he returned thither in 1851. In 1856 he was married, at Hillingdon church, near Uxbridge, England, to Louisa, youngest daughter of Henry Fisher, Esq., of Hillhead, Dunkeld. In the same year he removed to Kingston, the head office of the bank, and resided there till 1854, when, having become interested in a large lumbering concern in the County of Peterborough, he retired from the bank, and has since devoted himself to that business. In 1874 having sold the mills and timber limits, he removed from Peterborough to Toronto, where he now resides, and carried on the business of the Muskoka Mill and Lumber Company. When residing in Montreal, the Oregan dispute appearing likely to lead to a war with the United States, he obtained a commission in the Montreal Light Infantry, commanded by Col. Dyde. On leaving the province he retired with the rank of captain. He is connected with various public companies as a director or president of the board, is also a member of the Board of Trustees of the University and U.C. Colleges; but has not aspired to other public office. In 1840, whilst residing in England, he was initiated in the mysteries of Freemasonry. He served as W.M. of St. Paul's Lodge, Montreal, and J. in the R.A. Chapter of the same lodge. He has since youth attended the services of the Church of England, which he believes to be the best form of church polity, as a monarchy is the most perfect form of civil government. He has always held what are called evangelical views, and is a member of the diocesan and provincial Synods. He was one of the founders of the Church Association, organized for the purpose of putting a check on ritualism, and now takes a warm interest in Wycliffe College. His political views have always been strongly conservative, but he would at any moment sacrifice party to honest and pure government. He has three sons, the eldest being a barrister in Toronto, a silver medalist of the Univerisity, and a gold medalist of the Law Society; the second a B.A., Toronto, studying medicine at Edinburgh University; and the third, also a B.A. Toronto, is manager of the Muskoka Mills, on the Georgian Bay. He has also three daughters.

___________________

Other possible references for the Maynes:
----------------------------------------

"The Maynes of Powis & Logie" a pamphlet by Alexander Morrison (A Learmouth & Son Stirling 1927)

"The House of Mayne" by Rev Robert H Main (published privately 1939).
__________________

There is only one marriage of a Frances Brown on Scotlands People and that is to a William Wilson on 7 Feb 1795.  Similarly for Family Search.
_______________

Family Search has baptism entries for apparently 19 children of Frances and John.  But of these 9 are duplicate entries, leaving only 10 separate births.  Of these 10 births, two reuse the names given for earlier births who probably had infant deaths.  Anyhow the 10 separate baptisms were:

1. Jane Mayne Campbell, born 6 Sept 1797, bapt at Edinburgh, d.inf?
2. Jane Mayne Campbell, born 3 March 1800, bapt at Edinburgh
3. John George Campbell, born 15 April 1801, bapt at Edinburgh, d. inf?
4. Mary Ann Campbell, born 9 May 1802, bapt at Edinburgh
5. Elizabeth Campbell, born 17 July 1803, bapt at Edinburgh
6. William Hector Campbell, born 29 May 1906, bapt at Dunipace, Stirling
7. John George Campbell, born 12 Dec 1808, bapt at Dunipace, Stirling
8. Helen Campbell, born 16 Sep 1813, bapt at Edinburgh
9. Alexander James Campbell, born 3 April 1815, baptised at Edinburgh
10. Archibald Hamilton Campbell, born 12 August 1919, bapt at Dunipace, Stirling
_________________

The following baptisms are on Scotland’s People
                                                                       Parish
Surname   Forename         Parents/ Other Details    Gender   Date     Number   Ref    Parish  
CAMPBELL  JOHN             JOHN CAMPBELL/FRANCES BROWN  M  24/10/1795  685/1  390 157  Edinburgh
                              FR9958 (FR9958)
CAMPBELL  JANE MAYNE       JOHN CAMPBELL/FRANCES BROWN  F  06/09/1797  685/1  390 247  Edinburgh
                              FR3 (FR3)
CAMPBELL  JANE MAYNE       JOHN CAMPBELL/FRANCIS        F  03/03/1800  685/1  390 353  Edinburgh

The next two entries  are not theirs but of a separate John C and Frances Forrester:

CAMPBELL  JOHN             JOHN CAMPBELL/FRANCIS        M  03/08/1800  622/   40 171   Barony
CAMPBELL  JOHN GEORGE      JOHN CAMPBELL/FRANCIS        M  15/04/1801  685/1  400 38   Edinburgh

CAMPBELL  MARY ANN         JOHN CAMPBELL/FRANCES BROUN  F  09/05/1802  685/1  400 0    Edinburgh
                              FRMR4-29
CAMPBELL  ELIZABETH        JOHN CAMPBELL/FRANCIS BROWN  F  17/07/1803  685/1  400 163  Edinburgh
CAMPBELL  WILLIAM HECTOR   JOHN CAMPBELL/FRANCES BROWN  M  29/05/1806  478/   10 221   Dunipace
                              FR185 (FR185)
CAMPBELL  WILLIAM HECTOR   JOHN CAMPBELL/FRANCES BROUN  M  30/05/1806  685/1  400 332  Edinburgh
                               FR249 (FR249)
CAMPBELL  JOHN GEORGE      JOHN CAMPBELL/FRANCIS BROWN  M  12/12/1808  478/   10 212   Dunipace
CAMPBELL  HELEN            JOHN CAMPBELL/FRANCES BROWN  F  16/09/1813  685/1  410 322  Edinburgh
                                FR446 (FR446)
CAMPBELL  ALEXANDER JAMES  JOHN CAMPBELL/FRANCES BROWN  M  03/04/1815  685/1  420 38   Edinburgh
                                 FR504 (FR504)
CAMPBELL  ARCHIBALD HAMIL  JOHN CAMPBELL/FRANCES BROWN  M  12/08/1819  478/   10 221   Dunipace
CAMPBELL  ARCHIBALD HAMIL  JOHN CAMPBELL/FRANCIS        M  12/08/1819  478/   10 231   Dunipace

Note that 8 and 9 are duplicates with two baptisms in two separate places.  13 and 14 are duplicates in the same place.

Then note that the 3, the second Jane Mayne, and 4, the second John, are a mere five months apart.  One or the other, or even both of them, is not a child to this couple.  I think I need to get copies of the Register page for both of them to see if anything further can be found.
__________________________

Some of the above entries re-use the name given in an earlier birth.  This not uncommon practice indicates that the earlier birth was followed by an infant death, so they can be looked for now:

                   Birth               Death possibilities
   Name         Year   Place        Year               Place
1. John         1795   Edinburgh    1796, 1798, 1799   Only the Edinburgh ones
2. Jane Mayne   1797   Edinburgh                       None in Edinburgh
5. John George  1801   Edinburgh                       None full stop

That was a good idea while it lasted!
_________________________

Now to look at the five month pregnancy:

The entry for the baptism of the second Jean Mayne reads:

“p.350  for Edinburgh on the 18th March 1800
“. . . .
“John Campbell Tertius Esqre W.S. and Mrs Francis Brown his spouse -
“St Andrew Church Parish a Daughtere Born 3d Current Named Jane Mayne.
“Witnesses Messrs: Harry Alison and Peter Couper both Writers in Edinr.

That’s pretty conclusive.
_________________________

And for the second John it reads:

“John Campbell and Francis Forrester . . .

So that is pretty conclusive too, this was NOT a child of John C and Frances Brown.
_________________________

This raises questions over the two John George births, both with no wife’s surname.  I will waste some money to see if either Frances is a Brown.

The John George born in 1801 is also to John Campbell weaver and Francis Forrester, so not a Brown descendant.

The John George born in 1808 definitely had Frances Brown for his mother.
_________________________

This leaves Elizabeth born in 1803 and no clear indication on who her mother was:

“P.160    Edinburgh 15th August 1803
“. . . .
“John Campbell Tertius W.S. and Mrs Frances Brown his spouse St
“Andrew Church Parish a Daughter Born 17th July last named Elizh.
“Witnesses. Messrs. John Hepburn and Misr.     both Writers in Edinburgh
________________________

This leaves ten birth that are now known to be born to either a Frances Brown or with a second name of Mayne to cement the Brown ancestry.
________________________

In 1847, her death:

Parish:        St Cuthbert’s, Edinburgh
page NO:       290

Deeath date:   6 January 1847
Name:          Mrs Frances Brown,
Retationship:  Souse of
Spouse:        John Campbell Esq WS
Age:          
Death place:   21 Union St
Death cause:   Influenza

he Monument to the Cambells at St Cuthbert’s church, Edinburgh has this on it:

               JOHN CAMPBELL WRITER
     TO THE SIGNET NORTH CASTLE STREET — 1790

                FRANCES ALLEN BROWN
        WIFE OF JOHN CAMPBELL OF CARBROOK
                DIED 1847, AGED 69

            JOHN CAMPBELL OF CARBROOK
                DIED 1855, AGED 84

I am not convinced there is any connection between the two John Campbells; it could have saved them a few bob to borrow some of the unused portion of the memorial stone.
__________________

Their marriage index on Scotlands people:

Surname   Forename  Spouse Name                 Age   Date     Parish Number  Ref     Parish
CAMPBELL  JOHN      FRANCES BROWN/FR2234 (FR2234)   23/12/1794   685/1      520 183   Edinburgh
______________

From an image of the parish register and transcribed by me:

[                       Edinburgh           Ref 685/1        ]

“(178)                  Edinburgh 19th December 1794
“. . . .
“23rd
“John Campbell Esqr of Annfield Writer to the Signet of Andrew Kirk parish & Miss
“Frances Brown North East parish Glasgow daugr: of Mr John Brown Mercht in Glasgow
_______________