Since all the ICs have had their markings removed, I'm not sure what the chips actually are. But, a quick check of the footprint of some 8-pin PICs verifies that I can swap the unknown chip on the PCB out for one that I can program.
I did a parameter search for an 8 pin device with PWM and an EEPROM, since I wanted to use hardware PWM and be able to save settings. Interestingly, the PIC16F18313 was the only candidate.
This PIC has only two PWMs, so I can't use it with all four LED drivers. I'll just power the white LED driver with this hardware PWM, and bit bang the remaining three 2 kHz RGB drivers.
A really nice feature of this modern PIC is the preferal pin select, which allows me to programmatically assign functions to pins. This way, I can assign the hardware PWM output to match the existing footprint. The other pins will just be GPIO, so nothing special needed there.