Unfortunately not, the other supply is a cheap nasty 350w thing that only has 20 & 4 pin atx power connections.
Just had a chat with our seniors electrician, he thinks it does sound like a faulty capacitor that's struggling to hold power, with it being completely drained when left and then shutting down at random times. Though he said the only real way to tel is a visual on the caps, which I can't do due to warranty.
Currently trying to source a psu to test with, something in the 600w region should be enough to get the basics running. This has to be the most frustrating fault I've ever had.