With so much focus on the power needs of a hair dryer I would recommend checking out new products. Prices vary widely for hair dryers and some are better than others. Dyson has one that is $400 which find is a lot to pay but may need less power to run. Point is you may find it is cheaper to get a different hair dryer which uses less power to operate. I recommend looking at the entire picture, to make the best decisions.
Sitting here with a large bald spot on my head with thinning hair, hair dryers are not a concern of mine.
I used a Xantrex prosine 2000w inverter charger with a 100 amp charger output on my prior service truck that worked very well for me. I don't remember what it cost but it was a pure sine wave output and recharging the batteries was fast. For most uses including a hair dryer a cheaper modified sine wave model with 40 amp charger built in for about $500 might be a good choice.
I have looked into the use of inverter/chargers extensively to reduce generator runtime. I found that the weight of the batteries, cost of replacement and physical size was too much for mobile use. A quiet generator set was a much more practical solution along with a small inverter for satellite and tv. Currently I use a 12v flat screen tv in my camper and don't need the inverter at all. In the cooler months I use a single Honda 2000is which weighs 46lbs. (less than a deep cycle battery). In the hottest months where AC is needed I use a Honda 3000is which with outboard fuel tank weighs in about 180lbs. That is still a lot less weight than setting up an inverter with batteries.