Also, I see little point in inverting up to mains only to run a heating load - you would be better off running the heating load direct from the battery / source assuming that it is close enough so cabling size & losses are sensible.
The only point I can see to inverting up is that an inverter plus mains powered heater might be cheaper than a more-specialized low voltage heater. Are there cheap low-voltage heaters, perhaps made for trucks, boats or caravans? A standard wind-turbine dump load would do the job, of course. Avoiding the inverter would help reliability.
If the inverter is in the house so that its losses contribute to heating then overall they're not really losses. The same applies, of course, with cabling losses for low voltage.