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Welcome, I'm Chris (aka Moose).
Not much I can say about the excessive delay in getting a response is I am sorry. I am willing to try and help if help is still needed. Just reply and let me know.
I am baffled that in 5 weeks they have not been able to drive the vehicle and diagnose at the same time to see what readings are going out of whack and causing the shut down. You can electrically monitor all sensors with a DVOM or scan tool while driven as well as pressure test the system while driving. Quite often techs hook up the tools for monitoring and attach the tool to a wiper arm so it can be seen then go driving. They appear to not be thinking outside the box on this one and putting forth that special effort, there is not much to monitor for the AC system. First off they need to confirm the PCM is not sending the ground signal to the compressor relay and that is to blame. If so then they need to monitor the sensor inputs the PCM needs to turn on the clutch. Those include the thermal sensor, temp sensor, and HVAC control.
Now that I have posted a partial or complete answer, you will see you can rate me. The deposit you have already paid is not transferred to me until you rate me. I would like to receive nothing less than 5 star excellent ratings. So if we are not there yet, please reply and let me help more. If you're not pleased, let me know because rating bad does not result in a refund, there are better options. Rating now will not close this post nor prevent replying. Thanks Chris
Best of luck.