Why will your passport renewal take longer than normal? renewing in Bangkok can usually be done in less than 3 weeks, sometimes it's as quick as 10 days. You keep your old passport while they're processing the new one so you can still get Extensions in country, although you're not supposed to be able to travel overseas on it.
The only risk in going down the overstay route for kids, is should you want to send them to school in Thailand they may need clean passports and if they will be living in Thailand long-term it makes sense for them to be legal. Otherwise overstay!