Sonds like an intermitent voltage drop to me... I'd jump +12V to the ECU power pin, and drive like that. I can't tell you what intermittent ECU shutdown drives like, but if not that, I'd take that +12V wire and run a few other things. Also, try driving with the 02 disconnected. Could be MAF, so once warm disconnect and drive around to see if it does it. (will drive very slowly and badly, so take a lot of care doing this). One can jumper a lot of things, but if u jumper the AFM, only use 5V. Not all things are 12V
Typical culprits are body electronics modules (under the rear seat) and fusible links (tho they often look and test ok).
And a left of field suggestion.. Is there a dodgy car alarm in there trying to disable the ignition?