“Gillis might be a shortened form of that” or it could be a shortened form of the Yiddish “gillisse” (balded) as indeed the actor playing Dobie was, so as to hide with a razor-close crew cut his dark hair, in a part calling for a blonde. It’s less of a leap than from Julius to “Groucho” Marx.
Continuing on with this, perhaps the “razor-close crew cut his dark hair” (i.e. Dobie Gillis) was actually short for Gillette razors and Dobie adopted a short version of it as his stage name.
I think my post is a cut above the rest, said “Sweeney Todd”.