I tried letting you know in my reply here that the issue you are facing is mostly likely an issue with the power supply to your flash chip, not the programmer.
Have you tried programming the flash chip with the mainboard powered? If you do try, make sure to not connect the VCC pin (pin 8) from your programmer/power supply to the chip.
From the datasheets I could find, the flash chips for the Framework 12 and 16 are basically identical other than the voltage (the Framework 12 uses 3.3V instead of 1.8V). Datasheet for W25R256JVxx (Framework 12) & datasheet for W25R256JWxx (Framework 13 & 16)