How do you deal with an extremely uncooperative shard member?

The correct answer is you don't own the shard, CG does.

Shard mafias are a violation of the ToS for the game but coordination is done out of the game, making punishment non-viable. Shard Mafias are a gray area in the rules where they are a violation of the language surrounding manipulation of competitive game elements but CG cannot levy in-game evidence to ban people for them because that evidence exists in Discord.

You shouldn't be doing it, but most shards have them so the community in general thinks they are a thing.

You solve this by changing your perspective. It isn't YOUR shard, it's the shard you and your Mafia happen to be in. You don't get to set rules and enforce them.