Sorry, I have to go a little against the grain here.
I don't care what anyone else does, but for me personally, it may not exactly feel like "cheating" (after all, you're not really getting anything much out of it), but it does feel like lying to wear a button that says "today is my birthday", when "today" isn't really my birthday. But that's me. Everyone should do what they feel comfortable with. Meaning, if I were you I'd let my kids decide whether they think it's right to wear it or not.
We were there a couple of weeks ago, and dd got the birthday button on her birthday, wore it the one day, and then took it off her pin lanyard the next day. She also refused to join with all the birthday folks when they were called up on stage at the Spirit of Aloha dinner show, because it wasn't her birthday yet, even though they had invited everyone with a birthday in that month, and hers was the very next day, albeit the next month. Those are my dd's standards and I'm not going to push her to do something she doesn't feel right about.
In fact, that reminds me of a girl we saw in the elevator wearing a birthday button the day after my dd's birthday. I said something about yesterday having my my dd's birthday and the little girl started saying that hers was in a couple of weeks, but her mom or grandma (not sure) cut her off and said "yes, today is her birthday!" The girl looked really embarrassed.
So anyway, just trying to say, offer it to them if you want, but don't insist if they aren't comfortable with it.