If your heat doesn't work, you might have a problem with the car's cooling system. Is it that the blower doesn't turn on, or it never gets hot?
If it doesn't get hot, at least make sure (when the car is cold!) that there is enough antifreeze in it. If that's the case, its an issue of the mechanism that diverts coolant into the heater core for your heat that isn't working. Could be the control unit or the switching valve, but it narrows the issue.
And you definitely can't run a space heater from an inverter. Even if the inverter is 100% efficient (it isnt), a 750W space heater would draw ~60 amps @ 12 VDC. You don't have a circuit coming into the car that can handle that. We're talking at least 4 gauge wire to handle that on a 12V system.
Yeah, the coolant level seems fine. I did a flush a few months ago when I had a recuring "low engine coolant" indicator. I followed the prestone label exactly, except my radiator didn't have a draincock so I drained it by unclamping the lowest radiator hose. I filled back up with a 70/30 mix.
I am not an expert mechanic, this is about as technical as I get.
BTW the low engine coolant indicator still kept coming on and off occasionally until about a month ago it went away and I don't think I've seen it since.
Heres a better description of the issue:
The heat will generally blow cold air. But occasionally hot air will come out. I usually leave the fan in the off position and when I feel warm air radiating from the vents I turn the fan over to the 1 or 2 position. The air usually quickly turns to cold. Then I turn the fan back off.
Also, the heat seems to be more consistent, hotter, and last a lot longer before going back to cold if I am at sustained highway speeds. 80MPH is a lot better than 70MPH. Also, the heat works much better in 40 degree weather than 20 degree weather.