No weapons is quite hard ( Tried it after reading your comment, got beaten by Remy two times lol. ) though I don't think you need any mushrooms to win. The magical mushroom wand is often more than enough once you pick up the bow.
btw, do you need to go to the shop to farm mushrooms? I think after you get the bow you can go back to the surface and spam attack the mushroom summoning rod from the other side without getting knocked back.