If you're plugged into shore power, you should have more than enough current capability to run a hair dryer in any of the electrical outlets. We do it all time (well Ginnie does since I don't have enough hair to use a hair dryer on...).
Actually, you may throw a breaker, depending on what else is plugged in, on that circuit . The beauty of breakers is that you can unplug the extra device, flip the breaker, and you are back in business.
Just to be mindful that the outlet in the bath is on the GFCI circuit which includes the outlets on the kitchen island and the storage and docking station. Anything else on that circuit when using the hair dryer may cause a breaker trip.