The two options that make the most sense:
Have your electric motor running with 700-800rpm idle speed for the compressor and connect it with a belt. This would be slightly more inefficient if you have a lot of city traffic, but you could also add an start/stop automatic as this modern car probably has. Start/stop will stop the air conditioning equally in ICEs and in EVs.
The other option would be to use an electric compressor. While you said that you didn't want that, it might be a simple option for you:
In my understanding, the compressor is toggled by an electromagnetic relay that connects the compressor to the fan belt/motor by some sort of clutch (it may be hydraulic).
So it should be possible to use a relay to connect your high voltage circuit to the battery when you need it.
The rest (air mixing and so on) will be done by the air conditioning system itself. If I'm wrong and the car actually has an effect on the intensity of the compressor, you should still be able to simulate that the same way the car would do it.
If you really want to use a separate motor, you'll have to go for a motor up to 7-8kW since the compressor can use up to 10% of the cars maximum power. I'm not sure if you have the space for such a motor.