Well, there's a somewhat obvious solution here.
Everyone in the auction "levels up"
We all start as slave.
When a player reaches 3 cards bid, they may ban 3 elements from bidding on them. They may not ban anyone who has bid on them already.
When a player reaches 6 cards bin, they may ban an additional 3 elements. They may not ban the element that is currently buying them, or the element that bids on them most frequently (most number of bids, not necessarily most cards bid)
At 12 cards bid, the player may ban a third set of three. They may not ban the current bid winner.
At 24, the player picks his team.
No backroom deals, no accidentally putting themselves in a room too high, and no all out bidding sprees of 24 on one player.