I think the existing compressors in conventional split air conditioners can be used for variable speed operation as well. The motors in the convectional compressors are permanent split capacitor motor.
Are you absolutely sure about that ?
There are two very different types of split phase motors.
The first type is the capacitor start and run type in which the second out of phase winding is always connected via a capacitor.
These have very limited starting torque and are usually only used to drive fans, blowers, and similar loads which can easily start and gently accelerate up to speed with loads that require minimal low rpm starting torque.
The second type is the capacitor start motor.
Here we have a massive start winding which is used to produce a starting torque often multiples of the full load rated running torque.
The start winding is disengaged with a centrifugal switch once the motor reaches full rated speed, which it will do in typically about one second under a full starting load.
These motors are used for hydraulic pumps, cranes, refrigeration compressors and similar applications, which require a huge initial starting torque.
A typical sealed refrigeration compressor will usually (?) be of the capacitor start type, and when running, only has a single winding connected. The start winding will quickly overheat and burn out if run continuously, its just not made to survive that.
You cannot soft start a refrigeration compressor, it will quickly pump up to full high side pressure and simply stall the motor. It has to hit hard to reach full rated speed very quickly, before the high side pressure can rise up high enough to stall it.
Induction motors just cannot generate high enough torque when run much below full rated speed with full mains frequency applied. So for refrigeration service you need a very violent quick start up.
You will find these motors rated for "starts per hour" because the start winding needs to cool back down. As soon as you try to speed control it, the start winding will kick in via the centrifugal switch, and the motor will not last long running like that, even in a refrigeration cooled sealed unit.
Three phase motors are ideal for speed control, they can deliver 100% of full torque right down to zero speed, provided the fan cooled motor is cooled externally by a continuously running air blower.
It should work in a sealed unit too, but I would be a bit cautions about that.
You can soft start it even for refrigeration service and it will work fine with a normal variable frequency drive.
That is the only practical way to do it. If what you are suggesting were possible, there would be a lot of single phase variable frequency drives on the market for all kinds of applications.
As it stands, split phase motors can be speed controlled, by phase control, (voltage only) but only if the load is of the fan or a blower type, where the torque requirement falls away very fast with speed reduction (inverse cubed law usually).