If you still want to attempt to get it figured on your own, I will try to help where possible.
There are some ways you can monitor power to modules by back probing a small light into the power feed of a module, like for example the HVAC module, but it's not as easily accessed, unless you disassemble some of the dash to gain access. So, when the bulb is wired into the power circuit, you can see that it's receiving power by the bulb being lit. If while driving the bulb were to go out and it was wired into the power feed to the HVAC module, then you know that you just lost power to the module and you might find that it's unresponsive now.
In my previous response I pointed out that the HVAC nodule receives power from the BCM through Fuse F46 10 A on connector C228A pin 26. You could back probe carefully with a small pin into the back side of C228A at pin 26 and have that supply power to the small light you temporarily wired into the circuit and you would supply a ground to make the circuit complete. When you start the vehicle, that pin should now be supplied power and should stay that way until you turn the car off. You said that the HVAC can stop working while driving. With the bulb wired in and on, if you see the bulb go out, then the BCM just lost it's supplied BCM power. Hope this makes sense.
This doesn't condemn anything just yet, but it now tells you that something either in the BCM just opened the circuit or there is another issue causing it to do so.
Basically, the BCM like other modules, has power and ground, possibly a few of each, like a Hot At All Times and switched power from say an input like the ignition, then a few network connections as in MS CAN and HS CAN, followed by inputs and outputs. The BCM has a lot of control, but it relies on multiple other modules to do work too.
As Rod indicated, if you can read the networks modules live while your driving with Forscan than you can see what happens when the HVAC goes dead.
Interestingly, the HVAC panel with dual automatic temperature control, as many know, can have issues with the touch capacitive switches going nuts and driving the HVAC system into odd loops of operation, like full hot, full cold, etc.