Thanks everyone for the input. You've correctly shifted my focus from just the conductor size to the bigger picture of permits and the inspector's final say.
@PGB1's point about the AHJ is my main concern now.
Before I move forward, has anyone here actually had an inspector sign off on a similar large compressor install in a small machine shop? I'm trying to figure out if this typically gets approved or if it often gets stopped by main panel or service limitations, regardless of the wire size used for the new circuit.
I'm not sure what you are asking above.
If the existing service and equipment are suitable for the existing loads, plus the new compressor load- the rules allow it.
I've installed hundreds of step-up transformers for equipment (including air compressors) over the decades. Same for delta to wye, etc. It is a routine practice. (Be SURE to use a step-up transformer and NOT a step-down installed backwards. People learn this the hard way!)
Size the transformer and the wiring appropriately- remembering the inrush current of an air compressor can be quite large. The machine should have a tag stating the circuit size it requires. If no tag, ask the manufacturer for a copy of the manual & data plate.
The No Surprises Way:
You can always bring to the permit issuing authority details of your existing service and the load. Use the actual load and not a list of circuits breakers. Indicate what each load's job is. (Lighting, tool, convenience outlet, heater... and their load.) Most will be more than happy to review and offer guidance. Also ask if she will require the 125% rule for the new compressor.
And, I really want to emphasize the importance of a permit.
Even if a fire or injury isn't compressor or transformer related, the insurance company can get out of paying should there be a fire or injury.
I have been called as an impartial witness for tort actions and arbitration often.
Example: Someone runs a new circuit for a convenience outlet with no permit or inspection. Later, a roof leak gets a high bay light fixture wet and someone is shocked by that light fixture which was installed with a permit & inspection years ago. Nothing at all to do with each other. Insurance still gets out of paying for the injury and damage because the adjuster found one (unrelated) thing- the new outlet- that was not installed under a permit. Owner pays medical and fire damage out-of-pocket. Sometimes- medical forever.