We have this too, so am marking my place in the hope a wise person comes along with the answer.
It was hair wash night here tonight (I leave it as long as possible between washes) and it wasn't quite dry at bedtime, so I said we'd have to use the hair dryer.
She goes mad if I use a hand dryer in a loo, so I was expecting another meltdown, but she loved it! Next hair wash night I am going to reminder her that she only gets the hairdryer if she has her hair washed & see if that works.
My chiild is three. He is completely terrified of getting his hair washed. I attempted it tonight but could hardly get the shampoo out. He was screaming and shaking. I do not have a proper bath, just a small plastic one and a shower. I have tried the handbasin and have even resulted to taking him swimming so that at least his hair gets wet.