It's totally feasible, although you'll need some patience going from decent gear to great gear. The prices on good greens have definitely gone up - how quickly you can affordably fill those out will depend on server and some luck.
Heirlooms are competitive in most slots and BiS in some, and you can enchant those on a different server. A lot of BiS enchants on off-pieces are either cheap, or have cheap alternatives that are almost as good.
You can use a level 110 class trial to apply armor kits to BoEs (pretty sure you need BfA for them though). They're also great for herbalism/mining if you want to farm up mats to level professions. I think there's a 3 hour /played limit on class trials now.
You'll need a bit of gold. Selling pets is the best way to move gold across servers, but it can be a slow-moving market. BoA guild tabards (
https://www.wowhead.com/item=69210/renowned-guild-tabard) work well to get some quick gold if you have some to spare on another server. You resell them at a pretty big loss, but they can be worth it. The guild vendor in Shattrath is discounted by Sha'tar rep, so you can save a bit there.
Most classes gear pretty similarly, there won't be a huge difference between them. I guess a mail/leather caster using full fang set would need fewer BoEs.