Given your answer to Greg's question I would not twist the two black wires together. Ideally, if you have access to the wiring for the fan, take some detailed pictures. Sounds to me like the fan is wired with two hot wires. If you can't get to the fan: (Goes without saying, turn the power off at the circuit panel when you do this.) Disconnect the two black wires from the top of the combo and put wire nuts on the exposed ends. Turn the power back on and see if the fan works with the other switch. If the fan works then you are good to go. At least 1 if not both of the black wires that are disconnected from the combo are likely live. If you have an electrical tester you can determine if one or both is live. Keep them capped and put them back in the box when you install the GFCI outlet.

NEC 680.23 requires that underwater luminaires more than 15V must be GFCI protected. The "typical" pool light at 110/115V requires GFCI protection. In a wet niche, they must also be mounted a minimum of 18" below the water surface Low voltage pool lighting at 12V does not require GFCI protection. If the low voltage light is set in a wet niche installation, the niche must be bonded inside and on the back side. Nicheless, low voltage, LED lights do not require grounding or bonding. Nor do they have a means to do so. Depending on the manufacturer, they utilize either 1.5" or 2" PVC pipe as a means for installation through a pool wall. They can be set close to the water surface. Generally 8"-10" below the water surface provides the best illumination. ALL the pool equipment since 2008 has to be GFCI protected. Prior to that, it was just pool lights. Bonding requirements have changed also. BONDING IS NOT THE SAME AS GROUNDING. A qualified and experienced electrician is the best person to review your pool set up.

"Only the breaker would, if enough total current was passing." Well, sort of. Just to dispel any possible confusion: A GFI breaker implements two functions. The first is overcurrent protection. The breaker trips if the total current drawn from the circuit exceeds the breaker's rating. For branch circuits this is normally 15 or 20 amps. The more it exceeds the breaker's rating, the faster the breaker trips. The second is ground fault protection. The breaker trips if it detects that the current is asymmetrical across the power line. The amount of asymmetrical (ground fault) current that trips the GFI part of the breaker is MUCH lower than the current that trips the overcurrent part . (Confused yet?) To explain this second function: it's not 100% accurate to describe it this way, but it's easier (I hope) to understand if you think of the electricity as going out the hot side of the receptacle and coming back in the neutral side. The GFI monitors both of these. If the amount of current coming back is less than the amount going out, some of it must be "leaking" to ground. That leakage might be going through a person or animal, risking shock or electrocution. So when the GFI sees that, it instantly opens, shutting off the power. With most GFIs, an imbalance (fault current) of 5 milliamps (0.005 amps) is enough to cause a trip.

As Ron said, GFI breakers seldom fail. GFI receptacles are another story, they are notorious for failures. So it could be the GFI, but more likely the GFI is telling you that something is really wrong. Basic troubleshooting is in order. Ideally you'd measure the actual leakage current, but there's a simple if less accurate way. If the GFI never trip when nothing is plugged in, disconnect the charger from the extension cord and plug the cord into the outdoor receptacle. If the GFI trips with the cord plugged in but the charger disconnected, most likely it's the cable; replace it. A cable lying on the ground and/or in the sun for years will deteriorate and its insulation will fail. A much better and safer solution is to have a proper receptacle installed in the shed. If the cord alone doesn't cause tripping, plug in the charger. If the GFI now trips, have the charger serviced or replace it. A charger stored in a protected outdoor area can get all kinds of junk inside it -- cobwebs and mouse excreta and/or nests for example. These can get damp and cause leakage current. If a mouse mama raised her babies in there, you'll probably decide to junk the charger. Cleaning up mouse urine and the resulting corrosion ain't no fun, and could be a health hazard.