You aren't holding the key to Start the whole time it remains running are you?
Does the OP on the gauge get up to normal cold range or stay extremely low?
If the fuel filters and the OP switch connections look good, I might tend toward swapping out the OP alarm switch( not sender) between engines, or possibly just testing it with wires disconnected (and starter circuit bypassing it) with an ohmmeter to see if it switches as the oil pressure briefly tries to pump up during the running. Not sure how well the latter might work.
But if that switch was bad I'd expect shutdown might be more abrupt.
I'd think if a fuel pump relay or ECM relay was bad (or their on-engine fuses) it likely wouldn't start at all.
You might also see if you can find a friend or tech with a MEFI- 4 diagnostic scanner like Rinda TechMate, MercScan or their Marine Diacom , or similar software like OBD Diagnostics' MEFI-Scan4, ... with the Volvo adapter and a MEFI diagnostic manual ( included with the Diacom.) That could reveal for instance whether and out-of-parameter Engine Coolant Temp (ECT) sensor was telling the ECM the engine was hot when it was cold, etc. . As Meide suggests above , a faulty sensor feeding contrary voltage input to the ECM can quickly throw fuel/air mix or timing out of running range. Or a perfectly good sensor sensing what it is supposed to sense that is bad.
Probably not related, but it might be worth mentioning again that at least 2 posters here have reported their quick-stalling issue ended up being caused by an internally-shorted hour-meter that was grounding out the ignition to stall out the engine anytime the hour-meter made its every-30 second advancement "click". That meant the engine would stall any time from 0-30 seconds after starting depending on when it next "clicked".