I found a much better piezo in my parts bin. It makes a world of difference, it's quite loud! The size was also perfect, I could just stick it to the battery with some double sided foam tape.

Song still sounds bad, but that's a story for another day.

After checking the datasheet, I'm driving the piezo with a current that exceeds specification by a bit (nearly 5mA vs 3mA). A resistor in series or a different piezo would last longer. It seems to be doing fine for now though.