Soldered the rest of the components onto the proto hat over the weekend. Used a short length of USB cable from the jack on the hat to the one on the Pi - had to really fuss with the length, and ended up making it almost too short, but it works. The jacks, wires, and components sit kind of high on top of the hat, but I think it will all fit inside the box once I cut the holes.
Changed some of the pin numbers in the code to match where I connected the necessary LCD pins and buttons, and added some code to use the Pi's built-in pull-down resistors. Turned it on and ... it's alive! I plugged in some headphones and the MPKMini2, and I got piano!
Next steps: cut holes in the box (+sweet paint job), make some tweaks to fluidsynth, and write really nice interface code for my python stompbox.