A square-wave inverter has a peak output voltage the same as the RMS voltage of a sine-wave so the power in a heater and in an incandescent light is the same. If you could somehow filter out the harmonics then the voltage will be too low.
An inductor and a capacitor to filter out harmonics above 50Hz will be huge and expensive. The inductor will have such a high resistance that hardly any current will flow.
Modern sine-wave inverters stepup the battery voltage with a small high frequency circuit and small transformer then use a high frequency pulse-width-modulation IC to make a switched waveform with many steps in it. Then a small LC filter removes the high frequency which smooths the steps into a sine-wave.