What are my options for avoiding the overstaying fee when I mistakenly overstayed my visa exemption in Thailand?

You have two (possibly three) choices

-Do nothing, get a 3 day overstay stamp in your passport and pay 1500baht

-go get a 30 day extension for 1900baht at the immigration office BUT don't get an overstay stamp in your passport

and the possible 3rd choice

-go to the border, stamp out of thailand, stamp in&out of another country and stamp back into thailand to get a NEW 30 day visa exempt entry stamp <- which you could do ONLY if you didn't get TWO 30 day visa exempt entries by land so far this year..

The fact you hold dual nationality doesn't mean anything at all :/

Good Luck, those are the only choices.

Just pay up and use your other passport next time.

Did you already get a 30 day extension on your current 30 day visa exempt entry?