I assume the washer isn't on it, only because I think you would know if you had to turn on the bathroom lights to use the washer.
What I would do is disconnect the wires on the switch and seperate the wire pairs of the two romex cables. You can turn power back on and detireman the hot black and white pair. That's your line pair. The dead pair should go to the GFCI. Turn power back off and disconnect the grounds.
At the GFCI disconnect the black and white from the GFCI line side. Also disconnect the grounds. Try to work the xx2 cable back and forth gently to determine if it is loose.
If it is loose, you can attach a XX/3 cable to the xx/2 and use it to pull the XX/3 thru to the other box. (The alternative approach is attach a pull string to the XX/2 and pull it thru, then attach string to XX/3 and pull it thru)
If it isn't, you are at a decision point. Do you restore things and live with it the way it was, attempt something else to get the new cable thru.
I think you need to realize that if the xx/2 is damaged, or if you loose it in the wall, etc, it may take opening the wall to fix it. Can you open that wall from the backside to avoid disturbing tile in the bathroom?
Assuming you get the XX/3 in place:
At the switch box, connect the line black, XX/3 black, and a black pigtail together. The pigtail connects to a switch screw. The xx/3 red connects to the other screw on the switch. Connect the two whites and reconnect the grounds.
At the GFCI box, disconnect the lights form the load side go the GFCI. Connect the lights black to the red from the XX/3. Connect the black xx/3 wire to GFCI line side. Connect the XX/3 white to the lights white and a white pigtail. The pigtail connects to the GFCI line neutral. Connect the grounds.
At this point, you can power on. The GFCI should be hot all the time, and the lights should work normally.