KK3 is Keyboard Model B008, between DX (B005) and DXG (B009). The Touch was next model. FW 3.4.3
I think that though the DXG is later than KK3, it's really just a DX with a better screen, thus likely not enough RAM etc to run later than 2.5.8 FW
Amazon don't even call the DXG (B009) with better screen a later gen, it's lumped with Gen2.
https://www.amazon.com/gp/help/custo...MQC26VQQMM8XSW
We have one Basic model in the family that's not a Paperwhite, but has better than Basic 167 dpi screen. Like a PW3 screen with no front light.