We're short on specific details here. Just to toss a few things out:
"Back-Feeding" anything (where you insert power into a place that's supposed to be an outlet for power) is never good, can can be downright dangerous. An example would be putting male plugs on both ends of an extension cord, to be able to power a dead circuit from a live one.
Your Inverter is 400 watts continuous duty. A portable heater is at least three times that.
We don't know if your Inverter is hard wired, or the circuits you mention are plugged into its outlets.
We don't now if there is a source of 120 Volts AC near where your Inverter is located. If the Inverter is near your Load Center (120 AC Breaker Panel, probably with 12 DC Fuse Panel), AND the Breaker Panel has an unused slot, you could install a 15 Amp GFI Breaker and run Romex to a Box that replaces your Inverter. You'd have the same outlets, and they'd still be GFI protected.
If all you have near the Inverter is 12 DC, then a new Inverter's easiest, but the $100 one mentioned above is NOT GFI and isn't meant to be hard wired.
I don't know if it'd be possible to plug a box with a GFI into that $100 Inverter, then connect your outlets to that GFI.