I don't love Billy, but I do think it's cute. Wilhelm & Willem both work as ways to get to Billy, but I only care for Willem.
I can't think of alternatives that haven't already been suggested other than bird names. Birds have bills, after all. Peregrine, Kestrel, Swift, Hawk? If you don't care whether or not people outside the family "get" the nickname, I think something like Peregrine, nn Billy, could work.