I had one where the AC compressor clutch would stick but doesn't sound like your issue.
Now, when I plug the computer back in. I am able to use the A/C and AUC buttons. The green LED lights up! However, after starting the car, it takes about 10 seconds before the buttons no longer respond. Although the buttons can be pushed in the beginning. The A/C compressor won't respond.
Does the green light still go on and off?
Did you try and remove the fuse and see if it shuts off?
There is also an outside temperature sensor and a heater core temperature sensor that could be faulty adding to the fun.
Welcome to the forum!
95 E34 530I V2.37
===========
Those who make peaceful revolution impossible will make violent revolution inevitable.
John F. Kennedy