As mentioned earlier, sadly I am no expert. But I may have some info that is of use to you. So I'll give you what I've got. Expressed as what I'd do/think if I was in your position. Until you at least hear from someone more knowledgeable;
Module
BEM has no DTC function;
This is normal for AU Falcon.
The
BEM module doesn't have standard continuous monitoring system (that FORScan reads DTC codes from). Diagnostic codes for
BEM module are displayed by HEC/IC module (instrument cluster).
C1781 Module IC Instrument Cluster Engine coolant temperature signal missing / fault;
Does the IC gauge seem to work? Could be a problem with the sensor unit or something between there and the IC? Or could be just be a false positive caused by your main problem? I'd get my main problem identified/solved first and this might go away.
B2682 EEC (
PCM) Communication fault;
I'd believe this is highlighting your problem.
Either your
Engine Control Unit/Powertrain Control Module (ECU/PCM/EEC/ECM) or Body Electronics Module/Body Control Module (BEM/BCM) needs to be replaced.
As you said you have already successfully reprogrammed the keys ok using forscan.
You have a AUIII Ute. I believe all the
BEM's are pink and the same across all AU2&3 vehicles? If so just grab one of them cheap, swap it and use forscan again, to link it with your
ECU and
reprogram the keys again.
If it is an auto, 6 cyl and not an XR then you can probably also pick up a ute
ECU cheap at the same time, just in case that is your problem.
There are lists of the
ECU codes you can use around, or I can paste the relevant ones here from my list if you need them.