Hi,
I am running off grid. Need to power a 110V 13.5 BTU split A/C unit, (3000 starting watts and 2000 run watts is my guess)
If that split A/C is inverter drive, then no surge and 3000W as you suggest or even 2000W starting.
Otherwise ...
If 2000 watts by the nameplate, I would guess 10,000 starting watts.
I've measured smaller units, with a scope and current probe so I could see individual AC cycles. A meter isn't fast enough.
It was a small 400W window unit. Under mild conditions (not cooling a room on a hot day), running wattage was 1/2 of rating on label. Starting wattage for 0.16 seconds was 5x rating.
Some inverters quote surge rating, but for about 20 milliseconds which is not enough. Some quote surge rating for multiple seconds which is good.
Many offer a surge 2x continuous rating, some 3x.
If your A/C has VA (volts x amps) of 2000W on the label, expect 10,000W surge and a quality 5000W inverter is probably need to start it.
So as you say, "5-6KW inverter is adequate". Consider specs for SI-6048 (my picture)
11,000W surge for 3 seconds.
At 2000W or 33% of rated power, 95% efficient.
Larger inverter is better of course, efficiency and no-load power is the only concern.
Different brands vary about 4:1 in no-load power. But, the more efficient one generally cost more. This can be offset with more PV and more battery, also a cost, so include those in the dollar comparison.
You may be able to find a nice 120/240V unit that meets your needs so you have more flexibility. But you'll have to be sure of how much surge is available at 120V.