Dave "UTZ@aol.com" has produced on soc.gen.med two further generations of Belmeis ancestors but with no sources...
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John Ravilious posted a short pedigree from Philip de Beumes to soc.gen.med on 8 Jan 2006:

From Wrottesley’s “Pedigrees from the Plea Rolls”, p. 479:

Curia Regis Roll.  Hillary.  7. Hen. 3.

Salop--Roger de la Zuche sued the Abbot of Shrewsbury for the advowson of the church of Twange (Tong).
                          Philip de Beumes
                          seized temp H.1.
                                  |
     _____________________________|___________________________
    |                           |                             |
   Philip                    Ralph                         Alice
   ob.s.p.                   ob.s.p.                          |
                                                              |
           ___________________________________________________|
          |                                   |
       William de la Zuche             Roger de la Zuche
       ob.s.p.                         the plaintiff.
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And on 2 Jul 2006 on soc.gen.med, Pat Junkin came up with some Belmeis charters which cast some doubts on the received Belmeis lines.  More information eagerly awaited!
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Azure, semée of billets sable, a lion rampant or.

The above blazon came from Fox-Davies Arorial Families 7th edition of 1929.

July 2017, TFPL: Curiously the arms as drawn in RCLP-L’s Quarterings had the billets as Or, not Sable.  As I believe the arms were retouched at some stage, it is just possible that they had originally been Sable.  So I tried to find other sources of the arms of Belmeis, Beaums, Beaumes, etc.

No records could be found for these arms in the British Dictionary of Heraldry, Medieval Ordinary, in any of the four volumes.

Fox-Davies’ “Armorial Families” had two candidates, Corbet and Fellowes.

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For Corbet, see Vol I, pp. 429-430, they have the following succession of quarters:

24: Zouche
25: Rohan
26: Brittany
27. Beaumes

(That the genealogy for quarters 25 and 26 is defective does not matter as the sequence serves to identify the line of the quarterings and the epople involved.)

The first three of these have identical arms to those on RCLP-L’s quartering.  But the frourth is very different:

  27. azure, a lion rampant within eight billets in orle or (for Beaumes),
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For Fellowes, see Vol I, pp. 655-656, they have the same succession, differently numbered of course, and conclude this sequence with the comment of ‘qrngs 4 to 30 being thos of Corbet, Vn, Sal,, 1623’.  The seqence has the same balzons as those of Corbet.

This takes us to the 1623 Shropshire Visitation.
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In the Visitation os Shropshire, 1623, pp. 132-144, the Zouche arms do indeed appear in the list of quarterings but not the srms thought also to be inherited with Zouche.

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Later in July 2017: I also found the Beaumes arms in J Bernard Burke’s “The Heraldic Register 1849-1850, pub 1850, p. 31 for the 25 Quarterings of Hibbert of Everton, co. Lancaster.  In these 25 quarters is this sequence:

7. Zouche,
8. Rohand.   } Unfortunately the same genealogical
9. Brittany, } error as foun for RCLP-L in 1927
10. Beaumes,
11. Quincy
12. Bellomont,
13. Mellent,
14. Gwadyr

The arms given for Beaumes (Pillip [sic] de) are:

  Az., semé of billets and a lion rampantor.

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To conclude this, all of Corbet, Fellows and Hibbert blazons have the Beaumes billets as Or, not Sable.  I will accept the College’s exemplification as a semée for their arrangement, corroborated by Bernard Burke, and await better evidence.
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College of Arms achievement: Azure, semée of billets Or, a lion rampant or.
Armorial Families' blazon says the billets are tinctured Sable.

RCLPL’s Achievement of 1927 by C of A: (34) Beaums: Norfolk 14/28, Norfolk 16/19b.