If the a/c compressor always engages when the car is restarted, I will bet money against the pushbutton panel. Your seeing the compressor RPM circuit trigger. It compares the compressor RPM against engine RPM. If they differ too much, the compressor is shutdown - until the engine is restarted.
There are a couple of possible causes.
1) Excessive compressor clutch gap. Should be 0.50mm
2) Grease/oil on compressor clutch surfaces.
3) Loose serpentine belt/failed belt tensioner.
4) Bad electrical connections between compressor RPM sensor and Klima relay.
5) Bad Klima relay.
6) Bad electrical connections between engine flywheel speed sensor and Klima relay.
7) Bad compressor RPM sensor.
These are more or less in order of ease of checking/correcting. If your guys haven't checked the clutch gap don't let them put in the pushbutton unit.
I know of at least one case where a $1 spray can of degreaser fixed the compressor cutout problem.
As a last resort, the pushbutton units can be purchased rebuit for about $150. Dunno if partsshop sells them, but there are several vendors. Don't pay too much.