Just find a 3 phase motor that would fit in the unit (with the same or greater HP rating) and any surplus VFD with compatible voltage and power ratings.
There's also a hack to run a single phase motor from a common VFD. Don't expect as much control range and you'll need a somewhat larger VFD (still small as far as VFDs go), but not having to replace the motor cuts costs.
And you can indeed make use of variable speed. For cooling, slow it down to enhance dehumidification or speed it up for better EER. For heating, it's generally best to run it such that the discharge air just feels warm, with a gradual ramp up as soon as the burners are lit so that it can purge the cold air out of the ducts. (That can take some trial and error to get it right.) For ventilation, you'll have to again use trial and error to find the right point.
BTW, multi speed motors (used conventionally) have an efficiency penalty as low speeds cause excessive slip and high speeds only use part of the winding. But with a VFD, there's a trick to using the entire winding. With the motor unloaded, run it on the highest speed and measure voltage from the lowest speed tap to common. Divide that by 60Hz and that's the value to use for the V/Hz parameter in the VFD. (You'll probably need, for example, a 208V VFD and a voltage doubler to run a 120V motor that way.) Also replace the run cap with a higher voltage version if needed (440V rating for a 120V motor should be good).