MCA of 22A means it will work on a 25A breaker. Or a 30A breaker. MOP means "not bigger than" 35A breaker (or fuse.) You don't NEED anything more than 25A as the first available size bigger than or equal to 22A.
The Mr. Cool install manual wire sizes are, frankly, ludicrous. Doubly ludicrous for an inverter-drive unit (no huge starting surge.) 10AWG copper should be plenty.
Technically, you are supposed to follow manufacturer instructions. In this case, I think you have a very reasonable argument that you are following the manufacturer wire size instructions you got by telephone or internet rather than the ones that appear to be a mis-print or mis-translation in the printed documents. 10AWG copper is the correct size for 25 or 30A circuits under NEC.
The disconnect does not need to be fused at all. It can be any size equal to or greater than the circuit breaker feeding it. (I have 60A non-fused disconnects on 15A heat pump circuits - because they were the least costly ones in stock; and I see no particular value to having fuses in series with a breaker unless the breaker is itself a faulty fire hazard that should be replaced ala Federal Pacific / Zinsco...)