First question would be, do the fans run?  Try running the Dell diagnostics to verify (F12 at powerup). However, bear in mind that when you see this M1004 error, you're usually faced with a damaged system board that will need replacement.

Yeah, the fans pass diagnostics.  A bad mobo is the worst case scenario.  Is there anything else it might be, or is that pretty  much it?

One thing is that it always shuts down at the moment you click Exit to leave the BIOS.  I was hoping that maybe re-flashing might help.  Or it might make things worse if it shut down in the middle of the update.

I was also thinking of taking out the CMOS Battery long enough to clear everything.

Tried cleaning the RAM, but it still conks out with either one of the DIMMs installed.  Tried re-applying thermal grease to the CPU, but it goes out after a couple of minutes, even with fans running and vents unblocked.  It's starting to look like the mobo.

I'm about convinced at this point that it's a motherboard issue.  The system never gets hot enough to be genuinely overheating.  Usually the fans don't even come on at all before it crashes.  But the fans do work, as confirmed by the diags.  Thermal paste re-applied, with no effect.  Unless somebody has anything I haven't tried, I'm almost ready to break it down for parts, since everything except the mobo seems fine.

I'm still holding out some small hopes that it might be a genuine overheating issue, just because when it shuts down, it won't come back up for a minute or so, suggesting that it's really cooling.  It's just that it shuts down so quickly, and the fan works, but never comes on.  The only way to test that would be to get another e6540 and try its CPU fan assembly in this one.

The only hope for a non-mobo issue is that the fan never seems to come on before the system shuts down.  So, maybe it really is overheating.  On the other hand, the fan does come on during diagnostics, and passes diags, which still suggests a mobo problem keeping it from sending the signal to start the fan.