Adding Transistor to Maintain Voltage Across Proprietary Switch
Well, I managed to scare up a P2N2222A transistor from an Arduino kit my girlfriend has, along with a couple of 10k resistors. I wired those into a +3.3v header I found on the board, through the switch, and into the base of the transistor. After all that, it seems to be working fine.

It was also working with when I tried a 220 ohm resistor, and also while running off 5v. I decided that 2 10k resistors and 3.3v was probably safer in the long term as long as it worked.

Here is the schematic for my completed circuit: