I think that if there are two movies out that make the name the butt of a running joke, it's probably not a good idea.
Meet the Parents and
Meet the Fockers are pretty well known movies, and it is a running jokes through both that the male protagonist is named
Gaylord. Even if the name in question were
Barry, if there is that much history and popular usage of teasing with it, it's going to automatically be harder to use for real. I'd find something else, difficult though it may be.