This might be him.
In 1871, as James A, he's living in Birmingham with wife Ann a daughter and some step children surname Newton.
In 1881 he's living at 93 Long St, Aston with family, occupation licensed victualler.
There's no marriage for a James Abraham Beard and Ann Newton but there is a marriage for Abraham Beard and Ann Newton registered Sep qtr 1870 King's Norton.
If that establishes he could be listed as Abraham then on familysearch there is an Abraham Beard listed born 18 Dec 1836, christened 28 Jun 1837 at St Martin's, parents William and Ellen.
Also in 1861 there is a James A Beard living in Birmingham with wife Mary Ann. Again there's no marriage for a James Abraham Beard and a Mary Ann but there is a marriage for an Abraham Beard and a Mary Ann Mariss registered Sep qtr 1859 Birmingham.
There are 2 christening for daughters of Abraham and Mary Ann Beard but it looks they both die young as does Mary Ann, there 3 deaths in Birmingham/Aston on freeBMD for Mary Ann Beard, including one in 1864 no age given (possible daughter) and another in 1868, age given as 27 which roughly corresponds to her agw given on 1861 census.