Plugging a simple ac-ac transformer (wall power pack) into an inexpensive UPS

Many, many factors weigh into the severity but fundamentally yes, you're correct.

The AC resistance of the coil (mostly due to skin effect but also proximity effects) may be a bit higher at the harmonics causing additional winding losses. I doubt this will be particularly significant since the frequencies are probably still quite low. The other source of loss is core loss mainly due to hysteresis loss and eddy current loss. The severity of this loss has a lot to do with the transformer core construction and is to estimate. Again, I suspect the effect might be low.

So to be sure you'd need to carefully inspect the voltage spectrum at the output of the UPS and carefully consider the specifications of the transformer. Or you could just measure it. My hunch is that there would be a small but noticeable difference in heat produced. Curious to know more.

Personally, I'd compare the temperature rise of the wall wart when plugged into the mains vs. the UPS. Yes, the "modified sine wave" from the UPS will almost certainly cause the transformer in the wall wart to run hotter, but if that only causes the case temperature to increase from, say, 50C to 60C, I probably wouldn't sweat it... Bad pun intended.