My wife hasn't used the inverter to power up her hair dryer, but we have used the microwave. Both use about the same amount of amps. The microwave is only on for a minute or less, generally 15 seconds or so. The micro was wired that way when the coach was built and I never changed it. But, we use the micro very sparingly on inverter power.
Hair dryers can take up to 1,500 watts, or about 12 amps at 120 volts. That is a lot of power. Now, if you have four (or more) house batteries and an inverter that can provide at least 15 amps at 120 volts, you'll have enough for a few minutes of hair drying time. Your wife is probably wanting to use it for several minutes or more, which could really draw down your batteries. But, then you'll have to recharge the batteries anyway, so why not just run the dryer off of the generator?
Good luck. (And, and you indicated, happy wife equals happy life.)