R302.7 Under-Stair Protection - does drywall need to be below stringers or just treads and risers?
No, you can't leave the center stringer exposed. That would leave a path for fire through the drywall. You must create a continuous break, with all seams taped. 

I can't see an inspector flagging either of your other ideas, but you should probably just ask first. You don't want to do the job twice.