I guess it depends on how you define ''inexpensive''. I though $3400 for my shiny new 7.5kW lathe spindle servo was pretty cheap.
There are VFDs that are able to run permanent magnet rotors. It might be a bit fiddley to set up. I
think the Automation Direct GS4 units can do this.
There are a number of references to this around the internet, here is one that seems to be reasonably well written
Control Servo motor with a variable frequency drive I would not expect servo drive positioning performance out of one.
It is common for servo motors to have 6, 8, or 10 poles, rather the the more common 2 or 4 pole found in most standard 3 phase motors, so this is something to be aware of when setting up the VFD. You will need to match the VFD to the motor specs for voltage and amps. I would expect the motor specs to be on the motor name plate, you may need to search the Mitsubishi documents for better motor specs..
Getting it to position for a tool change might be a bit tricky. I was never able to do an index even with encoder feedback with my original lathe spindle motor, the VFD control was just not tight enough. But I did not try DC injection in that attempt, and the VFD did have that available. It might be possible to turn the spindle at maybe 1 or 2 rpm and just simply stop it with a prox sensor signal when it reaches the tool change orientation, I would expect that to be pretty close.
I'm still unclear as to the existing motor power. Depending on the motor power required, maybe a Clearpath 4 HP motor would replace the existing spindle motor. Would not be my first choice, but could be an inexpensive alternative.