The car air conditioner will work normally for a while after it is started, and then it will always slip, that is, it will not work normally.

After reading your question carefully, it seems that there is still a key link you haven't mentioned, that is: electromagnetic pull-in switch, commonly known as "compressor clutch"

Talk about this battery pull-in switch: this coil should get an action control signal (this is a point you check) after receiving the instruction from ECU that "the air conditioner needs to work"; Then, the front relay of this coil should be connected to the power line of the suction coil (this is another point you need to detect); At this time, the compressor clutch should act, and the normal refrigeration work begins.

From the analysis of the phenomenon you described, it should be that there is something wrong with the electromagnetic pull-in switch itself: internal short circuit. After its temperature rises, the short circuit situation intensifies and it loses its due pull-in force. You should use substitution method, try it!