Consider a 12 volt hair dryer adn a Heavy duty 12 volt outlet.
Inverter needs to be in teh 2,000 watt range,, 4/0 or bigger cable to batteries no more than 3 feet away (but not in the same air space) and standard Romex from there to the sub panel and from the sub panel to the Dryer
If your inverter has breakers intgernal skip the sub panel
If you plan on plugging a cord directly into an outlet on the inverter then twelve gauge please.. not 14 or 16 go with 12
Any time the load wattage has a comma 12 ga or larger is indicated, even if 14 meets specs.. 12 simply works better.
My inverter has 10ga from the 2,000 watt inverter to the sub panel, and from there 12 to the microwave and 14 to other loads.