Looking at births from 1837 to 1848 for Clara/Clare/Clarabel with surnames vaguely resembling BEAUCHAMP on FreeBMD and GRO there are:
BEESON Clara, Mar qtr 1842 Amersham vol 6 p310, mmn MILES
(died Jun qtr 1842 Amersham vol 6 p205, age 0)
BEACHIM Clara, Jun qtr 1845 Frome vol 10 p405, mmn PENNYMAN
(probably BEACHIM Clara Frances married Dec qtr 1880 Barton Regis vol 6a p214)
BELSHAM Clara, Dec qtr 1845 St James Westminster vol 1 p77, mmn GRAVATT
(probably BELSHAM Clara married Sep qtr 1870 St Saviour vol 1d p233)
BELCHAM Clara Selina, Mar qtr 1847 Strand vol 1 p417, mmn EAST
(no further information yet found)
BIRCHAM Clara Margaretta, Sep qtr 1848 Erpingham vol 13 p104, mmn WEBB
(probably BIRCHAM Clara Margaret married Dec qtr 1879 Marylebone vol 1a p1042)
and some marriages:
Sep qtr 1856 Axbridge vol 5c p961 BEAUCHAMP Richard Williams on same page as LOTT Clara, BIRTILL Sam, SPRATT Hannah
(discounted - 1861 Census has Richard & Hannah BEACHAM, Samuel & Clara BIRTILL)
Sep qtr 1867 Clerkenwell vol 1b p991 BEAUCHAMP George Thomas on same page as BATEMAN Clara Elizabeth, CRESSWELL Charles, WAGGOT Jane
(no further information yet found, but 1871 Census has Charles and Jane CRESSWELL)
Sep qtr 1867 Windsor vol 2c p734 BEAUCHAMP Thomas on same page as WARNER Clara, CURTIS George, BUCKLAND Jane Martha
(discounted - 1861 Census has Thomas and Jane BEAUCHAMP)
So, nothing conclusive yet, but one birth and one marriage could be worth following up.
I'm off to bed - I'll have another look after the weekend if no-one's cracked it!
Philip