Most modern-day, hand-held, home hair dryers are rated around 1500
watts. Run it on 120 vac and its no problem.
To get the same hair-drying capability on a 12 vdc circuit would require 125 amps of current which would result in the cord being thicker than a great
set of battery jumper cables.
Since the bath outlet is only rated around 20 amps or so of DC current, rewiring would be necessary for care-free hair drying.
So, any 12 vdc hair dryer available will probably only be good for around 240 watts of hair-drying capablility.
FWIW, when I was a little kid with short hair, my mom's 500 watt hair seemed like a waste of time. I could stand in an open doorway and dry my hair just as quick.
I doubt you will be happy with a 12 vdc hair dryer.