I've paid for 3 sets of braces so far.
One had one phase. Another had two phases. We have insurance that covers $2500. With the one phase, we had like a $800 down payment, and like a $150 a month for 18 months. The two phase had a smaller downpayment and payments for 10 months. Then we had 6 months off and did it again, although the second half was cheaper than the first half.
It is late in the year. Can you put money in a flexible spending account for next year? It will be pre-tax dollars. Plus, you can pay for the braces early in the year, and the FSA will still be deducting all year long, so basically you can give yourself the loan.