I have wired several pieces of machinery over the years manufactured in Japan. Large 5 color printing presses, 40' long 100HP spindle milling machines. They all came configured for 3Φ 220V 60Hz operation. Each building had a different service configuration, it is important to measure the actual service voltage before selecting the the transformer.
When the equipment contains VFD's or servo drives and computer controls, you really need a grounded wye output on whatever transformer you select. It's not because you need a neutral. Its because you need a ground that is symmetrical with all three legs for equipment functionality and reliability. Most of the rest of the world only uses wye systems and takes that part for granted. So when they say 220 delta, it usually means they don't really know what a delta is, they have no experience with deltas in their homeland.
Here a delta output has no way to be symmetrically grounded, you either can do a center tapped delta grounding , a corner grounding, or ungrounded with a ground fault monitor. None of these choices are any good for equipment reliability with drives and electronic components. Corner grounding can be downright destructive as soon as you power it up.
Using a delta output coil and leaving it ungrounded, is against the NEC and leaves you with liability down the road, but can function, for a while. I recently disconnected a 600V Canadian paper shear that had been connected for years to an ungrounded delta coil, but it had no drives or electronics.
The way I handle these is to specify a delta input/ wye output transformer that has enough taps to adjust the output to very near the 220V requirement. This is why you need to know the actual site service voltage, and it can vary widely during the day depending on the local grid robustness and neighboring businesses that have heavy loading during certain times of the day.
In your case of 480V input and 220V output, I would select a 480V delta input and a 208V wye output that has enough taps to get the output voltage very near to your required 220V. Tap adjustments are normally only available on the input side. So to raise the output coil voltage, you need to use a lower primary tap voltage. Higher voltage in, (adjusting the input taps to a lower voltage than actual), means higher voltage out. This will drive the transformer closer to saturation. but a 5% adjustment shouldn’t be a problem.
Be sure to properly bond the output XO terminal to the building and transformer and run a EGC to the machine grounding terminal. On the compressor I would look for another source other than the 220V output, unless the manufacturer will approve of the higher voltage input.