I beat it! I will say that I didn't realize that the Count of Down takes knockback at the beginning of the fight, so my first few games threw me for a loop because I would knock it out of the arena and be unable to damage it. I did eventually beat it by going in a circle.
I will say though, that I didn't use any of the firerate upgrades. Given the already limited ammo, I didn't see a point since I was focused on making every shot count. I don't know if that's just me.