Modified sine wave power inverter ?

Curling irons pull very little power. My wife's is only 125 watts. With a load that small the batteries should be more than adequate. Most any size wiring would carry that load as well. I feel certain there is some electronic temperature sensing circuitry that does not like the MSW power.
A pure sine wave inverter will fix that problem. I suspect you would only need a 300 or so watt PSW inverter to run a curling iron.