On a photograph I am posting on a bulletin board, want to identify my daughter-in-law's brother by his relationship to me.

Wouldn't that be your sons brother-in-law?

"my daughter-in-law's brother"
Or our daughter-in-law's brother is about as good as you can get I believe.

Bothers/Sisters-in-law's children become nieces and nephews and maybe some cousins by marriage but the spouses of one's children don't go further in defined relationships- at least I can't find anything.

Assuming that your son's wife is known, something like "Susan's brother" would work, especially if Susan is in the photo too and she is identified.

I agree with Smearcase. I don't think there's an actual word, or designation, other than "my daughter-in-law's brother" or "my son's brother-in-law". There really is not familial relation to you, at least not one that can be expressed with a title.