This looks to be your main panel, in a main panel only neutral and ground are bonded. Electrical definitely looks to have been upgraded since the house was built in the 50's as I'm thinking that's a lates 80's early 90's ge panel.
You do have a neutral coming presumably from the pole (the heavy guage wire landing on the right neutral buss taped white) which does give you a solution.
Best option would be to get some black #10 thwn wire and add a fourth conductor to the pool run. It looks like it is in conduit and 20 ft should be doable. Can you visibly see the full run? You may have to connect Mike tape/pull rope to the existing wires, pull them out to the single gang pull box by the pool panel, add the 4th conductor and pull them back to the main.
You unfortunately do not have the room to do so. Your existing breakers in the main panel appear to be "tandem" or "slimline", a 2 pole gfci breaker will take the equivalent of 4 spaces in the main.
If you can pull the wire I would do that. The only other solution I see that could work would be, if you have room, mount another small sub panel next to your main, feed it from the current 30 amp pool breaker (bringing a neutral and ground into it). Put the gfci breaker in there then feed the pool sub from that.