Your deck is actually worse against priest compared to any form of miracle because having small minions gives them things to steal and damage their minions to draw from, and they can easily clear your low health minions.
Also no saps means belcher destroys you, and no fans means hunter can out minion you.
Belchers aren't the problem. The real threat is wild pyro combos. Adding in the enrage minion (edit: spiteful smith) makes it harder for them to clear board, almost impossible to kill a 4/6 minion with 5-6 mana.
The problem thereafter is just being able to put out enough damage to actually out damage their hero power. That is where assassin's blades come in. I just took down a priest cause I had 2 assassin's blades, I already removed both belchers from their deck. It was a race from there. I had no minion presence, but blade flurry made it so I could wreck their field, deal a lot of damage in the process, and I had a second assassin's blade to keep up the pressure.
Granted, I did make a mistake by over valuing my defias ringleader. Instead of just running him into the leftover slime, I used backstab to maximize damage. If I kept backstab, I could have backstabbed the cabal shadow priest and cleared board with blade flurry then and there, and gotten far ahead that I would have won easily. But even when I screwed that up, I still ended up winning due to the sheer amount of damage I can put out over several turns.
I've also actually found that if I play around snake trap well, my winrate against hunters is phenomenal. I have enough direct damage spells to generally avoid triggering snake trap and that was really the main situation where I used fan of knives anyway.