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.
Don't think of it as a 120V receptacle on a 240V circuit. Think of it as a 240V load on a multi-wire branch circuit.
That is a legitimate wiring method. MWBCs can also supply 240V-only loads.
There's one gotcha. You can't have any receptacles on a circuit where the hardwired loads account for more than 50% of the total circuit ampacity. That doesn't apply to cord-and-plug connected loads, so the workaround is to make the hardwired loads not hardwired anymore. Fit sockets, cords and plugs.
So you have an MWBC. If you employ a GFCI on an MWBC, you must do it with a 2-pole 120/240V GFCI breaker (or deadfront that is inline). You cannot use a European 2-wire RCBO, because all 3 wires, neutral included, must go through the GFCI's internal sensor together.
In any case, in a water related installation, you want the GFCI protection well before you get anywhere near the water. Because you want the wires protected also - if the wires get wet, they are protected, rather than electrifying the water and causing drownings.
Best Answer
GFCIs work by making sure current is equal in and out (i.e. that all current is accounted for). As such, all (normal) current must pass through a GFCI device: All hots and any neutral.
Normally one attaches the neutral wire to the neutral bar. The GFCI instructions are warning that you can't do the normal thing, and "the" neutral wire must go through the GFCI breaker (for detection purposes) just like all the hot(s).
Since your circuit does not employ a neutral wire, this advice is irrelevant to you.
When wiring this circuit, you may find it prudent to run a /3 cable so it can be used later for some other appliance which may want neutral. In that case, leave the full length on the neutral wires, but cap them off.
All that said, the GFCI breaker will have a "neutral pigtail" (literally looks like a pig's tail, it's all curled up) -- this is where the GFCI gets the supply-side neutral from. You will probably need to plug that into the neutral bar on the panel; many 240V GFCIs need it to power their own onboard electronics.
As with all GFCIs, hook up the GFCI itself and get it fully working before attaching anything to the load side. You will easily catch any such problem in this area.