Earlier today, I made what I think are the necessary changes to the code. I should be able to test everything out now, but I have yet to be able to actually program the Pro Mini. I've tried using the USBTinyISP (which worked fine on the Pro Micro) and the FTDI adapter, but avrdude always fails. I've even pulled the Pro Mini out of the circuit entirely in order to program it, and it still won't work. Probably serves me right for buying cheap knock-off parts from AliExpress.
At any rate, my latest code is pushed to the GitLab repo, so feel free to check it out.
Edit: Apparently, the WAVGAT chips don't work correctly with ISP-style programming (SPI interface) but they do work when you program them via serial using the bootloader. On the other hand, the bootloader doesn't seem to allow you to read the flash, so it can't verify whether or not the write succeeded. I'll be testing it out tomorrow to see if the flash actually took.
Edit 2: Nope. Even though avrdude claimed it succeeded in programming the chip, it doesn't seem to actually do anything when put back in the circuit. I asked the seller on AliExpress about it, and after he shared the same instructions with me that were in the item description, he said he'd refund the $2.82 it cost me for 2 of these. That still means I have 2 of these sitting around, apparently unusable.
Edit 3: I've even tried using LGT's LarduinoISP to program it, clamping directly onto the chip's SWC and SWD pins since they're not exposed on the board... and still, no dice. More discussion can be found at https://hackaday.io/page/6452-wavgat-avga328p