Another link for small Marine/RV electric on demand:
http://www.eemax.com/EX2412M
Let's see... Running some numbers.
Going with a 5 minute shower. That is 5 minutes at 2.4KW from the inverter. Assuming an 80% efficient inverter that is 250Amps @ 12V at the battery for 5 minutes. 21Amp Hours use from the battery bank. That is 1/2 to 1/3 the usable charge from a single 100 Amp Hour deep cycle battery at that discharge rate. The faster you discharge a battery, the smaller it's usable capacity. It is mainly due to internal resistance in the battery. A battery with a lower internal resistance is able to utilize more of its charge at high output current rates.
Now if the incoming water is real cold, then you may wish to heat it up twice, or if you want two showers. That would be 42 Amp Hours use. A bit harder on the battery bank. That is well over half and possibly all of the max usable capacity from a single 12V 100AHour deep cycle battery at that discharge rate. At that point, I'd say two + 100AHr batteries is an absolute minimum with 3 or 4 being much better, or plug the RV in, or use a generator.
Note, the water heater unit draws 20 Amps @ 120VAC. That is at or above the current many older campsites allow. If the circuit breaker has been tripped a number of times it may not allow you to draw a full 20 Amps even if it is rated for that, circuit breakers do wear out and are usually only rated for 50 to 100 overload trips. A 30 Amp hookup would be required. Also it's current draw could easily overload smaller generators if something like a microwave or air conditioner was also running at the same time.
Because the load is resistive, it should only need a 2.4kW inverter to power it. Bigger would be better so you aren't pushing the inverter at maximum capacity.
Given the current draw at the battery. I'd have at least 2 batteries if not 3 or 4. Also I'd seriously look into a 24VDC house battery bank. That would halve the current in the wires between the inverter and battery bank, but it has other issues. Adding a 24VDC alternator to the engine is one of them. On the flip side 24VDC input inverters are usually 5 to 10% more efficient than ones designed for 12VDC battery banks. Same goes for refrigerators designed to run on 24VDC.
Yeah, the numbers don't look good.