Just checking in on this before tomorrow, and on a whim I had a look for something and have one quick addendum: I've located what I believe to be the right William B. C. Stirling, born 1822-3 in Canada, in the 1870 U.S. Census, resident Ohio; here his wife is Nancy C., his two daughters, Florence G. and Louisa D. are five and three years old, respectively, and a son, William (rather than Robert; I don't know whether this is necessarily accurate [certainly given the abundant errors I've encountered in U.K. census records over the years], or if his parents changed his name subsequently, or even if this 'William' is a younger son and Robert perhaps already left the household?) was born earlier that year.
There's also a marriage record, dated 31 March 1859, at Clermont, Ohio, for William B. C. Stirling and Nancy C. Banister. Somewhat confusing matters is the birth record for a 'Willie Stirling', 17 Nov. 1869 (therefore presumably the 'William' as above), with the mother's name given as 'Nancy C. Pumpelly', presumably indicating a previous marriage for her.
The 1880 census has 'W. B. C. Stirling', a carriage builder, of Batavia, Clermont, Ohio, father's birthplace Scotland, mother's birthplace Maine, U,S.A., with daughters Florence and Dean (sic; presumably the middle name of Louisa?) and son William.
These details might make William Stirling easier to trace, and any offspring.
Forgive the disjointed nature of this post, but I thought it worthwhile to add some notes whilst the details were to hand, that I (or, of course, anyone else, should they wish) can follow up on tomorrow.