A power inverter can run a space heater if the power inverter is larger enough and you have enough batteries to supply the inverter with the needed power.
Space heaters use around 1,500 watts of power so you would need a power inverter of at least 1,5000 watts to run the space heater.
Although you would really want the power inverter to be able to supply and output at least 2000 watts of power for the best performance and safety.
Also you need to have the batteries charging enough with the sun or something because the space heater will drain those batteries really quick.
I have used an electric space heater with a power inverter that I had that was 2000 watts and it ran fine but I had a good battery supply and a good solar panel setup.
So it can be done but it requires enough batteries and solar panels and a large enough inverter for it to work but it will work.