Good point. Didn't think of that.
Well in that case, the 2nd placed team chooses the team for the 3rd place team to play.

Or something.

I think I should choose who plays whom. I can't see any flaw in this, nor any improvement possible on my idea, and all can be assured of my utter independence, even the cheating Whinos and Stains.

Your innovative logic is indisputable, and if your guarantee of equitable fixture selections is good enough for me then the RL should be suitably assured.