I went through an simplified the circuit quite a bit to include fewer jumpers, and it doesn't seem to be a wiring issue.
I also tried adding 1uF caps to the power rails, just to see, and that didn't affect anything.
Directly connecting Reset to the pi while the others were connected via the level shifter also didn't work.
Assuming that it is rise time of the level shifter that is causing the issue, like in that post you shared, how should I go about adding pull up resistors? Just connect them between HV/LV and the pins of the shifter?
It's times like this I wish I owned an oscope or logic analyzer. 
Thanks again for helping me troubleshoot.