First a simple phase converter will not work with 110 you must have a 220 volt single phase power connection to power the converter. There are two types of converters, the cheaper (about $175 to $200) static type that will run a 3 phase motor ok but it will only let it produce 2/3 of it's normal power and that don't work well for a compressor. The second type is a rotary type converter that uses a capacitor bank and another 3 phase motor of approximately equal power to the biggest motor it will be powering and these work quite well, they do however cost about as much as a good single phase compressor! The static type can be connected to a separate 3 phase "idler" motor sort of like the more expensive rotary types and then it will let the work motor produce near full power but the key words here are "near full" power so again not a good set-up for a compressor and it is going to be a lot of trouble and expense.
Another way would be a VFD frequency controller that could be powered from a 110 source but these too cost as much as a compressor and I doubt you could find one big enough powered by 110 to run a decent compressor.
Long story short, forget trying to run 3 phase equipment from a 110 power source! Also for what you are wanting to do you probably will not find any 110 powered compressor that will satisfactorily do what you mentioned so you really need to first get 220 run into that shop then get a 220 volt single phase compressor.