Money does suck. It hurts so much to have to make choices we know aren't what we would truly want, but we have to make.
My question is, have you either 1) shopped around (maybe another orthodontist would do the braces for less?) or 2) tried to see if your orthodontist would do a payment plan with you? If you used your FSA for the surgery and then had could pay for most of your braces, perhaps your orthodontist would be able to make a payment plan for the rest, which would be $1500. The monthly payment would depend on what you could afford, but it never hurts to ask. How long would you need to work to make that money? Maybe you could do some temp work just to earn that amount.
Also, depending on your credit, etc. you might be able to get a loan from a credit union. My credit union offers personal loans, and while the interest rate isn't great, it would probably be worth it for you to not have to return to work before you were ready.
One other idea is to explain the sitution to the orthodontist and ask if you can get the braces towards the end of the year, get billed for what your FSA will cover in this year, and then elect the remaining amount next year. If they don't go for payments, maybe they would work with it that way. However, this may not be OK since you say the operation needs to be done soon.
Whatever you choose, I hope it works out for you.