What kind of lawsuit would have the best chance of blowing the lid on Wikipedia's shady finances?

You'd have to sue them on Opposite Day, and convince all parties to agree to observe the holiday.

You'd have to prove that you, personally, were injured by Wikipedia's finances before you'd have a basis to file a lawsuit. Unless there's some injury and some damages, there's nothing for you to claim in a lawsuit.

Honestly, it likely wouldn't be a civil lawsuit that blows the lid off the Wikimedia Foundation's (note, "Wikipedia" doesn't really have any finances) shady financial governance. More likely, it would be a federal criminal investigation.

However, I could potentially envision a class-action suit brought to bear by perhaps a dozen people and corporations who have been libeled by content on Wikipedia, sought removal or "oversight" of this content, but were met only with "it's not our problem" from Mike Godwin and his ilk. At that point, the lawsuit might attempt to supersede the Section 230 protections that the Wikimedia Foundation mostly enjoys, and push for a point of "public nuisance" and win compensatory damages from the Foundation. The case itself would certainly be bolstered by bringing into evidence the self-dealing racket that (purportedly) is the Wikimedia Foundation for Jimmy Wales and his investors.