I think you’ll need to use the Logitech software (or maybe AutoHotkey or similar software) to map buttons (4 or 5) through 12, to keybd presses.
Some tips:
BMS mapping for the scroll-wheel (aka. mouse button 3) can be done in the cfg file:
set g_sScrollUpFunction "FOVDecrease"
set g_sScrollDownFunction "FOVIncrease"
set g_sMiddleButtonFunction "FOVToggle"
(Although nb. I think the behavior to move head position in 6-directions by press-and-hold scrollwheel, is hardcoded… if you don’t have head-tracking enabled.)
There is also an option for button 4:
set g_bMouseButton4TogglesClickablePit 1
I don’t know of any similar cfg options for buttons 5-12.