Looking at around 5 horsepower for compressor and around 460V 24A 11,040W. A 400V battery setup lol that would be quite the sight to see, I'm going to see if I can see what that looks like.
11kW is just about the continuous capability of two SI-6048, assuming 25 degree C temperature. With 3 connected for 3-phase, would be good.
But a 11kW motor likely draws 50kW on startup. SI-6048 spec says 11kW surge for 3 seconds. Even four wired 2s2p like I have would be 44kW, which might or might not be sufficient.
If you use a VFD, that can start a motor without drawing surge current.
A compressor has an unloader, but once it starts turning pressure builds back up. The VFD I have on a pump is programmed to ramp up very slowly. You would want to ramp up faster so motor is at full speed before pressure builds. Or, arrange for unloading to continue until it is at speed.
A lot of battery, yes. My $5000 AGM battery bank would run that motor for about an hour or so.
If you have PV coming in, that will keep it going after battery has started the motor. 11 kW from PV is not unreasonable. A single orientation of panels would only peak at that mid-day. A 20 kW (STC) array arranged as two 10 kW arrays oriented for morning and afternoon sun could sustain 11 kW for many hours.
There are other ways to run 3-phase motors, including rotary converters. I had a motor-generator welder with 3-phase motor. I hooked 24V batteries to the welding terminals to get it spinning, then connected 240VAC single (split) phase to two of the AC wires and it ran.
Here's a European model inverter that would do what you want.
60 kW. But it needs battery 575V to 1000V.
Information on the Sunny Tripower Storage 60 (no longer sold by SMA) ► discover the successor here!
www.sma.de
Smaller pulley on the compressor motor, maybe smaller motor as well. Do you really need the full CFM it is rated for?