In my experience wirewrapping boards, never use the hand wrappers. They don't wrap tight enough, and after a year or so you'll be bedeviled by random glitches. Don't get a battery one.
Get a plug in one. They wrap nice and tight enough to cold weld the wire to the pin. Besides, they're a joy to use. BZZT! and they're done.
As I have said elsewhere on HN, I can count on a small number of hands how many times I delt with someone who knew how to wirewrap and had decent tools wrapping a bad joint. That stuff went to space, and war, after all. Oh, sure...after 30-40 years those DEC Unibus backplane wraps sometimes got a little flaky depending on mechanical stress (as my VAX attests), but I think we can forgive that. It was the novices with a pencil wrapper and a couple of dozen wraps under their belt that gave it a bad rep.
Get a plug in one. They wrap nice and tight enough to cold weld the wire to the pin. Besides, they're a joy to use. BZZT! and they're done.