For clarity, I would default to rephrasing to "this is a campaign funded by dark money" or "he is running a campaign funded by lobbyist groups" (you can maybe just say lobbyists). But of your options, my impression is that it's grammatically correct to hyphenate both modifiers, as they're serving as an adjective to modify campaign. So "dark-money-funded" and "lobbyist-group-funded." In practice it probably wouldn't matter, and native speakers would disagree or not be sure of the rules either! :)

Such a good question. I would say: "This is a dark money funded campaign" -- no hyphens! In another context you could say "a well-funded campaign" with a hyphen, but I think the hyphen has to do with the word "well", because another example would be "a well-known composer". Otherwise maybe none? Curious what others think. So I would then also say: "He is running a lobbyist group funded campaign" (Or, "He is running a campaign funded by lobbyist groups")