You
probably can with a Micro-Air to keep the cold start (and hot weather) starting amps down. Discussion assumes you have a 4T unit (or smaller), with similar characteristics to mine... It might be close.
Given you have a 25A inverter, with some surge capacity, you want to keep usage around 20A (80%), technically 3 or 4 hours at most over 80%, so slightly lower over intervals might be OK. Again look up your HVAC, here's mine for example:
It's a Goodman 4T, the last two columns 4T (and 5T). Spec is 26.7A total, reality is somewhat lower as expected. "North America" could mean Death Valley, but I doubt you are going to get much worse than here in Phoenix in July:
4 Ton package unit, July 20th in Phoenix,
daytime temps > 115F. HVAC usage graph. Peaked at 4953W / 240v = 21A, granularity is 5min.
This is as bad as it gets. In August when the highest temps were around 110F it peaked around 4600W / 20A. Right now in April it's more like 3500W or around 15A. More modest climate I would expect maybe 4000-4200W, YMMV.
Assuming that is ALL you want to run, probably be just fine, depending on your HVAC. Ought to be a way to have it pass thru a 40/50A from the main panel for when the inverter is down via the grid lugs, use the 6000 as an ATS. You will want/need a minimal amount of battery to smooth the Solar, 100A (5KW) output may be a little light, I'd get two rackmounts, 10KWH, it may need to start (and run) the thing when the sun goes behind a cloud or something, the Micro Air helps tremendously but it still draws an awful lot all of a sudden on startup, that is the hurdle.
Get a meter/monitor on the line, see what you have. You might start with the Micro-Air installation, then swap the 50A breaker with a 40A on the panel you have. No tripping, you might be OK. Get a Square D or other HVAC rated breaker, most are these days anyway. I'd Wire the load lugs from the inverter thru a standalone 40A breaker to the compressor, and peel the 40A in the main panel over onto the grid lugs on the 6000. Then set the inverter for shutdown at 10-15% SOC or something, and let the grid take over.
Very doable IMNSHO, if the inverter can handle the starting surge.