First thing is to browse what code is already out there, these are the ones I found so far:

https://github.com/ataradov/usb-sniffer-lite

https://github.com/tana/pico_usb_sniffer

https://github.com/sekigon-gonnoc/Pico-PIO-USB

Especially that last project seems very versatile, if I end up needing more than simply switching the input (keyboard/mouse) packets to different pins on the Pico.