Thanks everyone for your responses. I played with the shifter a bit, and the starter would turn for just a split second as I shifted from neutral into reverse before it cut out. I jacked up the car and messed with the alignment of the switch until I found the right spot... which doesn't have the alignment points precisely lining up anymore. I still feel like I should have gone with the OEM switch, it might have saved me a lot of time.

Thanks again for all your help!