What I would do in your situation, I think, would be to find a gym description on Disney's site that mentions the $12 fee for non-guests, print that out and take it with me to that gym. I could be wrong, but I think they'd be less likely to turn you away if you had a copy of the policy from the web site.
If all else fails, stop at the Guest Services desk at your resort and ask for a gym recommendation. There are plenty of gyms in the surrounding area, and the guest services folks should have a list of gyms that offer day passes and aren't too far away by cab.