Generac 22kW has a 100A MCB at the generator. There is no reason to run 4/0 between the generator and the ATS. I use #3 from the ATS to the gen.
I happen to know for a fact that the rxsc/rxsw200A3 ATS have lugs that will fit at least 3/0, because I land them all the time.
If the 4/0 al doesn’t fit, there is plenty of room for a Polaris in the ATS.
22kW is probably my most common unit, and I install them on houses with 200A services all the time. I also install 18kW units on 200A services. Just because the house has a 200A service doesn’t mean it needs it, or draws 200 amps. A proper load Calc will tell you what it needs. Sometimes it will need a 38kW liquid cooled.