My Opponent & I each have one prize card left and no benched Pokemon, and my Active Pokemon has only 100 HP left. If I Knock Out their Defending Pokemon that has a Deluxe Bomb attached, does the game end before they get to place 12 damage counters on my Pokemon to take their last prize card?

No, they get to take a Prize card. Everything, including all effects, have to be resolved before the game is considered as finished.

Relevant cards

SV7_134