The internal battery can be problematic. For example replacing it in Nikon D1 series required swapping a whole circuit board, since the battery is soldered right into it. At least that is how Nikon serviced it, but after the spare parts became unavailable we had to open the camearas ourselves and replace just the battery with a soldering iron once the circuit board was on the table.
Despite F6 being contemporary with the D2 series, the clock battery is not in the vertical grip... because there's no vertical grip. Nikon service is the obvious answer, or a seasoned indie DSLR wizard.