What makes it a "custom" laptop? And why does that make it easy to get inside? I note they do have a
forum - though that does not mean it will be of any help.
Oddly, when I search on that
model number, I get to a "Sorry, this product is no longer available" page on the company's site, which then dims and does not allow me to look around.
Tried 3 different browsers.
I don't think it's a thermal issue, since it had issues booting from cold when I first got it.
It may not be a heat issue in the sense of poor cooling, but a failing device causing a partial short will cause an increase in current in a circuit which can cause near instant over heating of affected devices. Blasting a desk fan on the exposed motherboard may keep it running longer - but also risks the bad device going up in smoke.
Got an infrared thermometer? Not exactly a conventional troubleshooting technique but these can help pinpoint hotspots if there is a short somewhere. But if the problem is an intermittent open, it won't do much good.
I agree with the course of action suggested above. I would start by verifying the output voltage of the supply while it is under load running the notebook. So with it shutting down in 5 minutes, you will have to be quick.
Unfortunately, it is not likely you will be able to find a circuit diagram/schematic so you can follow circuits with your meter. So a visual inspection may be your only recourse and if you see any solder joints that don't look right, a quick touch with a soldering pencil may repair the joint. I would start at the notebooks charger connection. Those tend to receive a lot of abuse, but sadly, not well.
I would also run it with bare minimum connected devices. If two sticks of RAM, remove 1. You can remove the drives and it should boot through POST, then halt when it cannot find a boot disk, but it should keep running. This will eliminate a bad drive pulling excessive current as the cause.
While I am not a fan of automatically replacing TIM (thermal interface materials) because contrary to what many believe, it does not need to be regularly replaced and will last 10, 15 years or longer AS LONG AS the cured bond is never broken. But since this is a used system and you may not know what the previous owner did, I would pull the processor (assuming it is socket mounted), clean the mating surfaces and apply a fresh new, thin layer of TIM. It only takes a few clock cycles to go from cool to overheated and a 3.4GHz processor (in turbo mode), that's over 3 billion clock cycles per second.