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Are you getting any trouble codes in the PCM?You can get them even if the check engine light is not on.
Is your 2004 F150 the Heritage body style or the new body style?
Ford had 2 different F150s in 2004.
The PCM controls the AC clutch relays which supplies power to the AC compressor clutch. If the PCM sensors an engine concern electrically mike a miss fire or a bad sensor, it will turn the AC compressor off. Then when you disconnect the battery, you are clearing the memory in the PCM and that is why the AC starts to work till you turn the engine off and restart the engine.
Do you have the digital or manual AC controls?
Do you have the ability to check the AC module and the instrument cluster for trouble codes?
You could have another module telling the PCM to turn the AC system off also.
Below are just the trouble codes and symptom chart for the AC system.
The EATC(AC), instrument cluster, SJB(Dash Fuse Box) and the PCM itself.
Not really, to my knowledge.
Below is the pin point test I would follow at my dealership for this problem.