Fallen Priest is the fourth stage of Sports Night.
This is a low-level cheese strategy, based on Neku Neko's video. You basically need to finish the enemy base before Black Bun Bun appears.
My winning lineup is
Row 1:C. Macho Cat (16), C. Wall (20), C. Axe (1), Manic Macho Legs (20), C. Giraffe Cat (15) (=Crazy Hearts combo)
Row 2: Bombercat (27), A. Bahamut Cat (30), 3 unused.
First, send out Manic Macho Legs, Crazed Macho Cat, and Bombercat. (Order doesn't matter that much, but if it didn't work, you can try it again with a different order.) Do not level up your wallet.
Keep on sending Crazed Macho Cats. Send out another Bombercat when ready, but not a third one. Bombercat and M. Macho Legs are a very effective combination in dealing with the Doge Darks.
When your Legs/Bombercats progress about halfway or more, send a C. Giraffe Cat. My Giraffe is pretty low leveled so they are "OHKB" (one-hit knockbacked), but you can send it earlier if yours is stronger.
Now when you have enough money, send A. Bahamut. After that, you don't have to do anything if your cats are strong enough.
Note that this strategy is pretty time-sensitive. When I first tried it, the enemy base has about 5k health left so I know it can work with more precise timing.