Although I'll probably just use a joystick, I did spend some time designing a custom control pad as a modular component to use. I had rather mixed results. The feel of it was great, but electronically it's not as sound as it needs to be. I should just create a membrane similar to the original, but I tried to make it a digital potentiometer instead.

The main box design is shown with the pad being identical in size to the original.

The circuit board is simply 5 volts in and then an evenly divided drop across 16 pads to be read by a single analog pin. The idea being to have a small membrane over it that the disc pushes on and completes the circuit. It uses the same basic design as the single common keyboard shown here: https://ardupiclab.blogspot.com/2018/11/an-analog-keypad-for-arduino.html. I do need to add a capacitor to make sure the voltage is consistent, but this was mostly for testing since I'm still planning on using a standard joystick. I'd like to get a slide-control stick (like the 3DS has), but I haven't really found one available.

Here are the finished (rough) results. I accidentally grabbed the wrong sized resistor footprint on the pcb, and I didn't want to engrave another board, so I just mounted them up. You'll notice the connecting membrane to the out-pin is missing. That's because I didn't find a solution I liked, and figured I'd come back to it at a later date. It's been 4 months and I haven't come back to it, so I'm guessing it'll be a while.