I managed to get the Bluetooth module working, and improved the issue where the shape would jump around the screen by sending a check byte to the program set to 255. By then making sure that neither potentiometer sent 255, I could use that first number to make sure that the game was reading a “complete” message.
Upon further testing with my classmates, the overwhelming feedback was to make this more of an arcade experience than a controller experience.
So I’ll be tabling the Bluetooth function in favor of building a small arcade cabinet that has reactive elements (like colored lights that match the player, and maybe some servos moving shapes on the exterior).