Yes, you can always switch servers with whatever character, as long as you pay for each individual one. If you’re asking for a free transfer, (and I researched the BC Classic forums) it appears on August 9, they’re going to consolidate down the BC Servers, according to this Blue post, here:
They even provided a couple of source links regarding “free character transfers”:
https://us.battle.net/support/en/article/000032067
I can only assume, if you can’t get a free character transfer, then it’s probably not available for the category, you’re asking for.