Does the compressor operate by having the motor run all the time with an unloader turning off the air into the screw when it reaches pressure or does it turn the motor off when it reaches pressure?
If the motor operates all the time you will probably never have the compressor pull a starting load when the Sharp is pulling a lot of current so you may be okay.
If the compressor motor starts every time you have a bigger chance of problems; the compressor starting may give you a low voltage condition on the Sharp.
I would be tempted to risk it and put the $4k into tooling .
An open mind is a virtue...so long as all the common sense has not leaked out.