Ok, mystery solved. Turns out that the Pro Micro that was I was using didn't have the voltage jumper (SJ1) soldered, which meant that VCC was Vusb minus the regulator drop. My PC has a USB voltage of about 4.3v, and the keyboard was getting about 4.0v, which turns out to be too low. When I tried the setup on another machine with a higher USB voltage, the keyboard and converter worked fine. Once I jumpered SJ1, the keyboard and converter work on the PC with low USB voltage, since the keyboard now sees 4.3v.
Hasu, as requested, I have attached a screenshot from a scope capture of the 'K' key without the converter attached. This was from the keyboard side, so the only pull-ups are the 2K resistors in the keyboard.