Great game! The art style and music both work really great for this one. The overall mechanic is pretty solid, and the bullet hell aspect is well complemented by the need to manage the amount of fired bullets on the screen.
I kinda agree with Charlie Gray’s feedback on enemy shooting. It gets a little bit too hectic too fast. However, I don’t think you should entirely get rid of it. Your idea of having melee only enemies would work really well - if you had two types of enemies, with only one of them shooting, it would reduce the bullet clutter already.
Another idea is to telegraph the shots - add either a short charging animation, or something like a progress bar on the shooting enemies. Maybe make them stop for a second before shooting. Something that would let the player know they are about to shoot.