How can I add a bathroom GFCI protected outlet?
If you provide power to the bidet, that circuit needs to be GFCI protected.  You can either run a line from the existing GFCI outlet (off the LOAD side, which is protected), or (I think) tap off an existing circuit that may be more convenient and put a GFCI outlet by the bidet.
When I added a washlette to my existing toilet, I chose to do the later as it was easier to tap into an existing circuit up in the attic above the bathroom and drop the cable down through the wall from the attic to near the toilet.