Hmmmm. That's not an area I've had to dig into yet on mine yet. I hope others with more experience can chime in.
1) I guess the easiest and first thing a person should do would be to disconnect the battery and let it sit unconnected overnight. You never know, it might chase away the bugs. That's been known to happen when wierd things like this occur. I know it sounds strange, but it costs nothing. Pull the fuse while the battery is unhooked inspect the fuse carefully ..and ohm it out, make sure no internal corrosion exists (or just replace it). As for the battery cables, clean the terminals and cable ends in the next morning before hooking back up b/c they probably need cleaning anyway. Make sure the small cable to the body is clean and making a good connection too.
If you can get to the BCM connector, might try wiggling that too, in case there's a faulty connection.
2) If you know someone who has a higher end snap on or Tech II scanner they could run the HVAC doors through various 'sweeps' to ensure the damper door motors are working properly and wiring is good to them and all readings and sensors are in accordance. And read any HVAC trouble codes from the HVAC module. If you can read the 'sequence of operation' of the HVAC system in the H3 Factory Service Manual, (below) it 'could' provide some clarity on what's NOT happening. There is a whole sequence of tests illustrated in the manual below to troubleshoot the system, but it takes a good scanner, like a Snap On or Tech-II to do them.
3) If that doesn't do the trick ....and you're the ultra-handy sort with a bit of a grasp on electronics (or willing to learn), you might carefully disassemble the HVAC controller where the dials are ...and look over the circuit board ..if there is one, for any corrosion, cold solder joints, leaked out electrolytic capacitors, etc. If it was me, that's what I would do before replacing anything.
4) If you don't want to take things to those levels, this might be one of those times where it's worth investing 1-2 hours of dealer diagnosis time. ??? My hunch is they're going to scan the HVAC and do the troubleshooting steps listed in the manual. They may have documentation on fixes that show what most fixes are. But if they fix it, might be worth it depending how much work you want to put into it.
https://schwarttzy.com/wp-content/uploads/2013/01/Hummer-H3-HVAC-1-4-1.pdf
Beyond that, I'm kind of out of ideas. Hope others can chime in. Good luck!