I think in general, Way of the Mouse sounds like the most Powerful Way. Sure there's plenty of weak and cards, but they're almost all strictly better than several of the other Ways.
Way of the Turtle sounds to me as decent contender. It also makes some actions super OP.
In many boards, Chameleon is incredibly powerful to the point of being occasionally broken.
I looked a couple examples and didn't find anything "broken" in the Chameleon. Sure it is a nice Way, but mostly it is like cool to have this, when there is a lack of draw/virtual money with abundance of the other or adds a flexibilty, when you're working with your engine.
Way of the Turtle at least gives you a Bridge combo. Simply buy Bridge(silver if not enough) and play it as turtle till you have enough Turtle'd Bridges too wipe a Province pile. And I thought about a couple other actions, which become great with a "start of turn" play.
While not always gamebreaking, there are several cards, such as Courtier that can function as superlabs or labs with upside, like Coven. Festival gets to be a better Lost City, Scholar gives you +$7, and Animal Fair can be a Council Room without the penalty.
Right, while not any more game-breaking than King's Court; it does make a lot of cards suddenly strictly better than various official cards.
And if you combine it with King's Court/Mastermind, it can really get insane - Scholar in particular, if you choose Chameleon for the first two plays and the normal option for the third, you have +$14 and a 7-card hand! If there's any +buy in the kingdom, you're almost guaranteed a double Province, and even triple Province isn't out of the question
And if you want to get absolutely absurd - Fortune with all that. Assuming there's enough +buys available, you could easily get quadruple or more Province!
Way of the Turtle with KC transforms your engine into something super-reliable: play and buy everything you need and then lay out your next turn any way you want to.
As I look at different actions it gives almost each and every one some new tricks.
It's downside that you'll need villages to play the cards reliably often, but it also turns the one that you would like to play at certain timings into much more deadlier ones - you can combine attacks into a brutal chain, set the trasher for when you'll hit the things you'll like to trash, wait out the Lighthouses, and simply "pseudo-trash" all unneeded action cards (like Cursed-out Witches, Trashed-Out Trashers and Ruins). You can make an interesting Turtle-Vineyard deck, with a little bit of support. And well it generally helps combine actions into some combos.
As for the Chameleon: I think you actually listed almost all more interesting cards. Courtier and Festival are very good. Scholar is not very interesting, it's too easy to hit province, which i think will make luck a dicisive factor (without KC/Throne combo). I looked through cards and Chameleon mostly makes better non-terminal silvers (as +1 Card>+1$ usually) and some attacks, that have normally given money become more brutal, as you can chain them like a Torturer.