I would suggest a reflux condenser above your tank like used on an alky still.
If your gone awhile, not using power, and your dumping to the tank allot you don't want it to boil dry. With a reflux condenser the boiled water steam will raise to leave the tank, it gets cooled to under boiling temps and condenses back to water running back into your tank. Since your running an open system, no pressure, I think you could maybe get by with a large pipe or copper tubbing packed loosely with stainless steel wool. Make it long so there is lots of length for cooling before the steam escapes the open end.
Also since water seeks it's own level, if your worried about running dry you could also have a second tank beside the heater to store extra water. This way your only heating what you need in the 12gal tank, not trying to heat too much all at once. When the level falls in the heater more water will flow in to replace it from the other second tank. Kept mounted at the same levels you could have all the water available you want, like 35-70gal but only have to heat 12gal at a time with the dumpload.
Since you want the hot water and making it with dumpload, cooling the tank itself or using a larger tank may not work that well wgen you are home using it.
Your part
"According to my calculations, It will take about three hours with everything at max warp to bring 12 gallons of water to a boil from 70 degrees, then about a gallon will be evaporated every 40 minutes. The system will not be pressurized so the water will not be replentished automatically."
The reflux condenser should reduce or stop that "gallon will be evaporated every 40 minutes" part. Extra storage in a side tank should not increase your heat up times.
If you figured correct, 3 hours is a long time to heat water when it's needed, and that's only if your batteries are full charged and your dumping power at full blast as you say, so I dought you would want a larger water tank which would take even more time to heat up more water.
If you have the room, I would lay a 35gal barrel on it's side beside the water heater then pipe from near the bottom of the barrel to the heater. Fill barrel to even with top of your water heater. Leave an air hole or mount an open fitting on top of barrel so water flows freely.
Not sure how you plan to fill your heater now, but this way you could also just fill the barrel as needed and always have extra water ready for use as needed.