Can I run custom firmware on a pre-certified RF module without needing to redo full FCC certification?
It has been my experience that you would have to re-certify if you are replacing the firmware in the module. 

Be aware that despite the pre-certified nature of certain modules that ultimately you or your company are responsible for meeting the legal requirements for emissions and immunity of the product as a whole. So even though a module may be pre-certified you still need to test your system with the module present. 

The only place I know of where a pre-certified type module may have some decided advantages are as follows:


For an embedded module you at least know that the pre-certified module is much less likely to be the source of issues for your system.
A module like an analogue telephone line modem that probably carries several pre-certs may excuse you from testing the analogue phone line requirements whereas the emissions and immunity part may need to still be qualified with your system as a whole. 
Externally applied attachments to your product such as power bricks and cords can come pre-certified and safety inspection labeled such that you do not have to deal with that over again for your product. (A very common reason so many products today have external power bricks or wall warts).