Switching GFCIs line-side tends to make them a bit cross
Plug-type GFCI devices revert to the "tripped" state when unpowered, unlike receptacle-type designs, which mechanically latch in both "tripped" and "untripped" states. This was done to allow them to protect against an open neutral, which'd otherwise defeat the GFCI protection and make it impossible for it to trip.
However, this makes them incompatible with being switched on the line side, as you see here. The solution is to move the GFCI protection to a location upstream of the timer -- GFCI breakers are ideal here, but a receptacle or deadfront GFCI can be used upstream of wherever the switch or timer is. Once you've done this, you'll have to remove the GFCI plug from the end of the plug-in pool's cord and replace it with a regular plug, with a big note on it saying "FOR USE IN GFCI PROTECTED OUTLETS ONLY".
Either that, or you can remove the plug and wire it into a timer that's then wired in series with another length of appropriate cordage that is terminated in the existing GFCI plug, effectively moving the existing GFCI upstream of the timer...
As well it should.
Normally, loads are connected between hot and neutral. Appliances are not supposed to connect hot or neutral to ground; ground is only a shield.
The GFCI compares current on the hot and neutral wires. They should be the same. If they are not, current has found another route, possibly through the grounding system (which isn't supposed to happen) and potentially through some poor human.
Circuit testers are trying to test whether ground is connected... cheaply. They mis-use "hot" as a power source, by connecting a light bulb between hot and ground. If ground is connected correctly, this will light.
In other words, it intentionally creates a hot-ground fault (by sticking a light bulb there). This is exactly the condition GFCIs are designed to detect.
I'm not talking about any ground-fault-test the tester may also have.
So why do testers often work? With a perfect GFCI, they wouldn't work. I suspect it is because GFCI's have detection thresholds above zero, and that is often enough for these testers to "get away with it". I even think there may be a tacit agreement among manufacturers for this, but obviously, lower sensitivity impinges safety. Remember a shock which only stuns you can kill you with secondary effects like falling or drowning.
So either your GFCI is pretty good, or your tester is pretty bad.
Best Answer
This sounds highly unusual some plug mounted gfci s need to be reset whenever they receive power but as you said you can unplug it and plug it in without having to reset it so I would check the wiring at the switch to make sure that it is breaking the hot wire and not the neutral also make sure the bare ground wire is not contacting the neutral at the outlet or in the switch box.