Some thoughts, and I'm happy to be shot down on any of them..
* Does your compressor never switch off - is it always either loaded or unloaded, but running? If true, this would seem to be one easy inefficiency to eliminate, limited by the number of starts permissible per hour possibly?
* Any change to the running speed will change the mechanical loadings on the compressor - the gas forces will remain similar but the inertial forces will change - can the design permit this? Not only bearings to consider here, I would suggest.
* A new motor, plus inverter drive is liable to be very expensive: plus, how much power is required by the inverter itself? How much saved money/increased efficiency is there likely to be in reality, and what will your payback be? You may find this is none/infinite - you may not.
* You are right to be concerned about vibrations at certain speed ranges. How are you going to find this out before purchasing and installing an expensive VFD system?
* The compressor cylinder valves will most likely be optimised to some degree or other for operation at the speed your compressor runs at currently. Will running at different speeds cause valve inefficiencies to outweigh any potential benefits of VFD?
* There are a multitude of possibilities for mechanical capacity control that would be a lot less capital intensive. Whether any of them would lead to any efficiency improvements for you (in the loosest sense) is a moot point, however.
* My opinion - do not do anything without a buy-in from the compressor manufacturer.