Fanuc does allow up to 8 BG logic programs, there is more info in the HandlingTool manual or the help button on the pendant.
Also you can use the "run" instruction from BG logic instead of the "call" instruction. The link below has some good info on how that works.
How to RUN FANUC Programs Concurrently with the RUN statement - ONE Robotics Company
In the past I have used something along the lines of this to monitor the status of a gripper while holding a part. The F[enableGripCheck] would need to be turned on after you pick up the part and turned off before you place the part.