I see nothing wrong with letting you pick from the full pool every time. I mean, there's no risk, right? Not like trainer lets you get high scores or pvp. If you want to test against a particular False God, you'll roll the dice repeatedly until you get the one you want.
So two buttons... one for random False Gods, so people can still roll the dice and see how well the deck performs in general, and a second button for specific False Gods. If there's one thing gamers hate, it's RNG, and while it has a place in the game, it really doesn't in trainer.