As it turns out, if a poll is public, you simply have to have a Patreon account, and don't have to be a current supporter of my Patreon.
Given that the poll's been up for a few days now, I don't intend to change anything, and to just let it continue. You at least have to be logged in as a Patron, so I don't see a way for people to use this to cheat the outcome, and if I'd known it worked that way, I might have considered it for something like this anyway.
While I don't consider this that huge a blunder (though it certainly is a blunder), I felt like I'd be hiding something if I didn't comment on it, so... Comment comment comment.