Actually one of my first thoughts was in fact to use a small series inductor as suggested, but when you think about it, when the scope is switched off so there is no load but the input to the SMPS is still connected to the inverter waveform, this inductor in combination with two of the diodes in the bridge rectifier of the scope SMPS would work to a certain extent as a voltage doubler boost converter. With the scope switched off and so no load on the power supply the input electro might get pumped up to way too high a voltage.
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As for power factor, a flat topped waveform as produced by the inverter is in fact an excellent waveshape for feeding a SMPS, at least one that doesn't have active PFC. Current is supplied for the whole length of the flat portion of the wave, not just the very tip as in a sine wave. Did a measurement with a Voltech power analyser years ago, sinewave was 0.6 PF, rectangular UPS waveform was 0.9 PF.