Don't worry about the refrigeration untill you can get it to stay on.
The compressor is controlled by the Klima relay and the V8 doesn't use a speed sensor on the compressor. As such about the only thing that should make the Klima ignore your "on" command would be a wrong temp reading.
I usually pull the Klima and jump the output to make the compressor run. With the Klima out the other inputs to it are available for testing. Get your wiring diagram.
Bosch Master, ASE Master, L1
33 years MB technician