All they've wanted to know from us is if we are celebrating a birthday. Well, if we want a button then YES we're celebrating a birthday, so it's not a big deal at all if it's not the exact day.
The buttons don't say "It's my birthday TODAY", they just say Happy Birthday, so there's nothing to worry about.
Well, actually, if the one with the birthday will feel the need to explain that it's not actually their birthday every time someone wishes them a HB, or feel uncomfortable with all the nice wishes they get, then don't get a button... But since it's you, I'm sure you'll be fine. (my son refused to wear his button even a day after his b'day, and I ended up feeling uncomfortable with the well wishes, but I've always had weird reactions to HB so I don't know what I was thinking by wearing the button LOL)