Since the difference in ups output and inverter output is of sine wave output and saw tooth output, its my personal opinion that it should not make any difference to feed an ups with saw tooth voltage. Wait for the opinion of some expert as its my assumption. I did use this sort of config for some time and my ups at times used to beep even with inv supply on and it used to switch off in 8-10 mins then. This behaviour was random and I could never figure out why did that happen. But no damage in my case ever.
To connect PC directly to inv: it may have some serious repurcussions... i never tried that.