TL;DR Safe answer = Hardwire
There are a few different pieces here:
GFCI affects stuff
That is, IMHO, almost entirely mythical. GFCI is, as I understand it, a basically passive system unless/until it trips. AFCI too. If any modern equipment is affected by simply being GFCI protected, there are major problems. And any older equipment - well, that is unlikely to be affected even by bigger things (think EMP effects - far worse the more modern the equipment).
GFCI is affected by stuff (i.e., False Positive)
That has some truth to it. Much more so for AFCI, as AFCI is a far more complex system. But even GFCI can be affected by legitimate minor current leakage. So there is some risk of False Positives with certain types of equipment. However, unless there is a chronic (e.g., every few days) problem, the only real issue is if there is a significant life safety concern. That concern is why in some places a dedicated refrigerator receptacle is exempt from GFCI, and why smoke alarms, fire control panels, etc. are normally not protected by GFCI or AFCI. The small risk of a serious fault being protected is outweighed by the risk of inoperable key equipment (safe food, fire alarms, etc.) not working.
What is most vulnerable to a ground fault?
Water. Guess what? A water heater has water. And it has electricity. That being said, a typical hardwired water heater doesn't need GFCI (at least as far as I know in most locations), so arguably even a plug-in water heater isn't really all that much of a concern. And actually, it isn't. In fact, I would argue that a gas water heater where the electricity is not used for a big heating element inside the water tank but rather for controls & ignition outside the water tank, the risk is really quite low. But the rules are the rules. In particular, if you make an exception for a plug in water heater, and then someone unplugs it and plugs in a tool that then has a ground fault, you've now killed somebody. Really. So the rules are set up to avoid the danger, even if it is relatively low.
Updated based on details in the installation manual, page 55. (Thank you NoSparksPlease for the link.)
I see only two practical options here. I really, really, really don't recommend cheating by swapping GFCI for a regular receptacle. That would violate code, though you might be able to get an exception from your AHJ, and there is always the concern that someone could reuse the non-GFCI receptacle for something else (I need to plug in my tools, I'll just unplug the water heater for an hour...) and have problems.
Just use it. If you get frequent (weekly or more frequent) nuisance False Positive trips then revisit the problem. If everything is installed properly, you shouldn't have any problems at all.
The manual does say "DO NOT connect to a GFCI or AFCI circuit." It gives no explanation. So doing this may be against manufacturer instructions, which by extension is contrary to normal code of "follow manufacturer instructions". I say "may", because I am not actually so sure about this. The instruction seems a bit strange. On the other hand, it is followed by the standard "don't use 3-prong to 2-prong adapters" and "don't use power strips".
Many appliances can be hardwired. There really is no reason (except if the manufacturer doesn't allow it) to not hardwire a water heater. After all, the gas line will be quite permanent, so why not have the electricity permanent too? Then, at least in most areas, you legitimately bypass the GFCI requirement.
In fact, according to the installation manual, the outdoor models are normally hardwired and the indoor models can be hardwired. Key point from the manual:
An ON/OFF switch must be provided and installed for the incoming 120 VAC power supply.
This can be a simple toggle switch (a.k.a. light switch). Since the units can share a circuit and use only 2 Amps each, you can wire up one switch to the panel (i.e., bypassing the GFCI receptacle, but could be daisy-chained from the LINE side of a GFCI/receptacle) and split the switched hot from the switch to go to both units.
Another key point:
A dedicated circuit is recommended for the water heater.
Note that this says "recommended", not "required". There is no real issue of overload - 4 Amps (2 units running at the same time) is much less than 1/2 of a 15 Amp circuit, so sharing the circuit with receptacles or lights is not a problem. Similar to sharing the ignition circuit for a gas cooktop with receptacles.