Camping Hair - Inverter or other ideas - Advice needed

Ughhh, yeah, no... :-p

I went through the same issues with the 12 volt hair dryers. I ended up getting this "hot air brush" which is only 250 watts. It doesn't work well on really wet hair, but if you towel dry it really well, it works! I use my Goal Zero 400 power pack, but an inverter would work, too.
https://www.amazon.com/gp/product/B000PY17PI/ref=oh_aui_search_detailpage?ie=UTF8&psc=1#Ask

There are butane versions of curling irons and curling brushes available too. Although, you probably won't find any until you get to a bigger center.