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I think the easiest way for Aquitaine to gain independance would be to claim the throne of Navarre or Aragon (since both those kingdoms are quite weak at the start). Declare war on one of them and France will almost certainly help you! In the peace deal, make sure you end up with a king title. The king of France can't be you're leige anymore (since you are now the same rank as him) so you'll just peacfully become independent.

Using this method, the King of France will actually help you break away from him! And you can still be friends afterwords (if you want to be). All the other methods of breaking away will almost always eventually lead to war.
 
However, if you time your war right, you can at least white peace out with him. I did, and am now on my second generation of the Kingdom of Aquitaine and Aragon. I actually just won my second war with France, gained one more province for me, and one more vassal.