Blackthorn is the dragon gym, correct? If you have any ice-types, that shouldn't be a problem at all.

[Elsa... would be lying if she said she didn't want to try out the gym thing whenever she and Anna made it to that city.]

Oh, right, you need some of those badges to use those HMs that allow you to travel easier, correct? It seems a bit odd that you need specific badges, rather than a certain number of them...