Well, one way to satisfy nearly all of those constraints, would be to implement (near-)full double elimination for the final-four.

So A&B play, and C&D play. B&D both lose, so they play each other, and the winners A&C play each other. Let's say B wins, and A wins. So then, B plays C, and the winner plays A in the gold/silver matchup.

For full double-elimination, A would get a second chance if they lost the final matchup, but that's neither needed nor practical here.

-ml