I believe there are problems you may not know about.
The inverter for one. Either have to leave it on unloaded, deal with "fast" cycling of the inverter and maybe huge surges to the caps, or do some fancy circuit work.
Small batteries with a 1000W load will have a large voltage drop very quickly, meaning faster cycling than one might expect. Surface charge will make it worse.
An reliable inverter able to handle the loads and cycling may (will?) cost more than 12V elements.
Together they make for quite a complex and expensive project.
Same end result would be DC elements, but cheaper, more reliable, and easier.
And there would be no wasted power in the inverter, meaning more heat in the water.
==
A small load (related to the battery bank size) could be run from an inverter pretty easy, with the hysteresis set wide.
However that's pretty much only for a 2 stage system. The first stage could not be large enough to act the only dump load.
Might be nice for doing something necessary with extra power (filling water tanks, couple 100W light bulbs for winter heat, 2nd small attic fan in summer?).
Maybe as an outlet that only works sometimes, for the laptop, cell phones, cordless razor, etc. chargers? That's pretty easy to do!
G-