I'm not aware of anyone that runs a 120v water heater through an inverter, for the reason you mention.
Most boats I've seen have a combination of 120v (for use at the dock) and engine coolant (a hose carries the hot "coolant" from the engine, through a heat exchanger in the heater, and back to the engine) in one unit.
There are also diesel-fired heaters (Webasto, Espar) which, in addition to heating the water, can heat the boat. We have one of those and have found it very handy. Compact, low power draw, and reliable. Some folks just go with a teakettle, or some sort of coil that sits on the stove or diesel heater.
The size you want will depend on your demands, but about 6 gallons seems to be common. With really good insulation you could have a larger tank and heat it up (by running the engine or the diesel heater) every 2nd or 3rd day instead of every day, but of course, that takes more space.
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