Using RPC (rotary phase converter) for a compressor may not be a very good idea unless you need the compressor for a short time and only occasionally. Otherwise, you'll have to keep RPC on for the entire time when you expect your compressor to switch on automatically as needed to keep its tank pressurized.
And since RPC is better to be rated, roughly, twice higher in HP to reliably serve a 5HP air compressor (which motor is always loaded), we're talking about 10HP RPC motor wasting power while the compressor is in a standby mode (=ready to add air to its tank as needed). Sure the idling RPC motor will not consume tremendous power, but it'll definitely consume more than you want to waste.
Although my home shop is on 15HP RPC, I wouldn't like to use it for my compressor which is free to start any time of the day to add pressure as needed. I turn my RPC on only when I need to use my other 3-ph machinery such as lathes, Bridgeport, surface grinder, buffers, sanders, etc. It may be 1-3 hours a day, but not 12-24 hours a day or a few days in a row when my compressor needs to be on standby.
So, a reasonable approach should be either to replace the 3-phase compressor motor with a single phase one, use a VFD or just sell the one you bought and get a 1-ph air compressor if you can find one in a decent condition and reasonably priced. Single phase compressor will be more expensive than a comparable 3-phase one simply because of higher demand (not many people have 3-phase power in their garages, and that's why you got yours for a pocket change).