The question is simple. If Skyrim won its independence, would it flourish as a independent kingdom or would it fall to internal strife or foreign powers such as the Aldmeri Dominion? Yes or no and why?
The elves would love if Skyrim broke away from the Empire, divide and conquer.
That didn't work out so well with Hammerfell, now did it?
The Aldmeri Dominion is a non-issue unless and until either Cyrodiil or Hammerfell are conquered first because until that happens the Dominion has no shared borders with Skyrim.
Skyrim wont prosper under Ulfric though. His hatred for the Empire will make any kind of favorable conciliation between the two impossible. He's going to have to deal with the loss of subsidies from the Empire and the resulting deficit in resources for Skyrim coupled with the damage the civil war inflicts on the province. Skyrim would suffer for many years under his rule if not longer.
If they invade by land they have to fight Hammerfell who has already beaten them and has ended their own civil war. If they invade by sea the Redguard navy will be waiting. There is no way the Empire will let them march through Cyrodiil.
If Ulfric is the issue he could always be removed now the the Dark Brotherhood is back.
The Empire isn't going to let an invasion sized Dominion army march through any portion of its lands. It's delusional to assume the contrary because if it allows that kind of blatant violation of its soverignty it may as well handover the keys to the Imperial Palace and White Gold Tower to the Dominion.I don't think that the Aldmeri Dominion would have many qualms of marching through northeast Cyrodiil or invading Skyrim by way of ships should the empire break up.