I've actually had a mixed experience with this, but I do not think they will make your 2 year old wear one. We were there last winter with our 18 month old and they specifically said she doesn't need one and never bothered even trying to put one on her. When we were there a few years ago with our then 2.5 year old twins, they did put wrist bands on them. One day we had to cut my daughter's band off before nap time because she was pitching a fit about it. I tossed it and never thought twice about it. When we got to the pool later that afternoon they had us go get a wrist band for her and told us the next time we have to take it off, to bring it back to the towel stand to get a replacement as they weren't supposed to issue any replacements without proof of the original band. They were not unfriendly or rude per se, just stating a rule. (I get the no replacement thing, but not as much the 2 year old needing it at all).
I can't really imagine making a fuss about a 2 year old wearing/not wearing a wrist band - surely they aren't coming to the resort to use the facilities on their own and a responsible party should be very close by and wearing the appropriate wrist band. I would not push the issue with your child and just make sure all is fine to not wear it when you request wrist bands/towels.