Since you're on a trip, I'd just leave the converter unplugged as much as you can. Your Air Conditioner doesn't use 12V and doesn't need the converter to operate.
Assuming your setup to charge when driving and don't use the lights much, you probably can make it a long time without the converter. I can easily make it 3 days without driving but I have LED lights. I've made it 9 days with AC hookups when I drive.
Change the converter when you get one, but that can be a big job. There isn't a rush for it. In a pinch, you could plug in a battery tender or other similar charger into the AC(the current) to charge your battery if you aren't charging while driving.