I have two passports, my child has two passports. Your daughter should leave/enter Thailand on her Thai passport. If you try to leave on a US passport with no entry stamps, immigration is going to have lots of questions. When you check in, you will need to show the US passport, if they see no entry stamp, they will ask for the Thai passport. I fly a lot, I always have to show check in both passports. If you have less than 6 months on the Thai passport, they probably will deny you boarding. You could try to argue w the check in that the US passport will get you into America, and that you don't need 6 months on the Thai passport to return, good luck with that. Or you can just get it renewed and not take the risk.
Also, I suggest renewing US passport in Thailand via mail. US passports renewal processing time is up to 3 months now in the states. International renewals are less impacted by delays. I did mine a few months ago and it took 5 weeks.