The Barnum Family, 1517-1904 refers to Israel as both Harrison Barnum and J. Harrison Barnum.
In the 1850 US Census for Carlton, Barry, Michigan the family of Israel H. Barnum was enumerated as follows:
Household #769; Family #788
Barnum, Harrison; 34; M; b. N York
Barnum, Eunice; 23; F; b. N York
Barnum, Elnora; 1; F; b. Mich
Hale, Simion; 6/12; M; b. Mich
They were living next door to the family of Hepsibah, the widow of his first cousin, Stephen Barnum, and to Mason Barnum, his first cousin once removed.