How can I fix flickering and continuous-clicking with a UPS on an inverter-supplied circuit?
What type of inverter is it? If it's a MSW (modified square wave) it's very unlikely that any UPS is going to be happy eating what it provides, since UPSes (even ones that generate MSW themselves) expect a real sine wave (or True Sine Wave - TSW) on the input side.



On the other and more relevant hand, you really don't need any sort of purity to feed most modern computers, many of which (check yours) are happy to take 100-250V at 50-60 Hz, (and don't care about TSW or MSW) since their own power supply is just converting that to DC with a switching converter. Only older linear-type power supplies are all that fussy about voltage and frequency. So viewing the  inverter as a huge UPS is probably a valid approach, and removes a complication. I would suggest a good surge supressor power strip for the computer.

If you really need (for some reason) the cost (initally and in replacing overpriced batteries) and hassle of a separate UPS, or equivalent, consider using an adequately sized "high quality inverter" that runs on batteries all the time, and a battery charger that can handle whatever your whole-house inverter throws at it to maintain charge on that battery all the time - essentially your own "dual-conversion" UPS (most actual Dual-conversion UPSes would have the same complaints about a low-quality inverter input as a normal UPS does.)

The other obvious solution is to plug the UPS directly into the Mains supply, so that the UPS works off its batteries when the Mains go out, and the inverter works from the inverter's batteries (or whatever powers it) separately.

If it is a MSW inverter, the likely reason you can make the UPS happy with other loads is that the other loads (particularly ones with inductance and capacitance) will tend to take the "sharp edges" off the waveform and make it better approximate a sine-wave.