sedfrey, it comes down to being very difficult to diagnose a problem with a laptop. The normal procedures we might suggest are useless because you can't disconnect various components and see if replacing this part or that one will work.
As said, see if it will work without the battery installed at all. If it does work and shows the same problems as usual, then it is eliminated as a possible problem. The laptop could be running mainly off the charged battery and when it runs low, it shuts off. The battery meter should aid you in discovering this.
The PSU in the case of a laptop is external - basically the AC adaptor. It may indeed be malfuntioning. You can test this with a multimeter, see if it's putting out the same voltage when the laptop runs properly as when it dies.