Well, after I thought I had it solved (it seemed the left-most two slots would cause it to flounder, regardless of which card was plugged in) it fell over again.

I put all the cards back in again, and ... had a weird thought. I put a small baffle in front of the intake fan, about halfway covering it (limiting the air flow into the case for cooling).

Without the baffle, it'll shut off in around 2 minutes. With the the baffle completely covering the fan, immediate shut-off. Half-covered... it's been running for two and a half hours on Low clock, no issues.

I'm... flabberghasted. I have to admit that the baffle idea came out of nowhere -- I thought that maybe letting it heat up a bit might help get it running reliably... pure voodoo on my part, I readily admit that. I don't like voodoo in electronics, so if this does indeed keep the silly thing stable, I'd like to figure out WHY. Is it because the fan has resistance and has to work harder? Is it because the cards get a chance to heat up a bit? Is something more or less conductive at the higher temperature, and thus "happier" from the perspective of a working system??

I'll keep monitoring and report back.