Maybe I could get a note from my wife dental office saying I'm getting my wisdom teeth pulled and need recoup time?
I don't advise this. You'd need lots of documents to go from the dentist to HR and back. One slip here and you'd be in a bad way. It's way more than just a note, if you even qualified for FMLA, and I believe that takes a year of employment first.
Just go to them and lay out the truth. For good measure, you can even offer to work a vacation week when you earn them. That will show them you're earnest. Most mgmt is good about this stuff with enough notice. Good luck.
If that fails, you could also consider an off work back injury. Near impossible to deny, and with no comp claim to pay on they're usually just happy you didn't get hurt on the clock.
All you'd need to do is keep them posted on your progress and get a return to work physical when you're all done. It's still risky as hell though because you'd be out of the country.