Ok that was hard!
At least I thought so. Probably the hardest first boss fight I've ever had. But damn if it wasn't satisfying. Took literally every single one of my bullets on one go. He died with my last shot to the head. I did spend a lot of time getting familiar with his dance routine though, but I could still never quite manage to get close to him reliably.
Needless to say, I was freaking out when I started running out of bullets and had no idea what I'd do when they ran out. And I didn't exactly plan on using the gun, but when I started my first "let's commit" run, I was then just like "aw what the hell, might as well just try the gun!". I'm SO glad it worked out on the first/only go for that...
By the way, I did get frustrated, and started searching on google a bit to see if there were nerfs, and people recommended getting that sword (last pic). I guess it is a random drop (?), but it was really funny to find it at the boss' chest.