Mulligans
Since in this tavern brawl your minions die at the end of your turn, you just need to play Mecha'thun with empty hand and deck to win the game. The minion summoned after Mecha’thun dies does not affect Mecha’thun’s effect, because deathrattle activates before it is summoned.
Stall through the whole game with all the healing and removal, cycle your deck, empty your hand and deck, then play Mecha'thun on empty board.
I tried to use Galvanizers in previos version but never managed to play Mecha’thun before turn 10 so I cut them.