The Studio15 is the wedge-shaped plastic laptop, right? Since it's plastic there's likely a bit of flex in the chassis (probably what caused it to fail in the first place) so you could try cold-booting whilst applying pressure in various directions to see if that makes a difference. If the short or gap is so close that a little bit of thermal expansion does the trick, you should be able to get a similar result to thermal expansion just with some flexing.
Ah, I should have thought of this! I've repeated it twice now, when leaving it on to heat it up to start it up, I just pressed my palm firmly against the heatsink/fan area, and it started doing the overheating beeping. Every time it gets to that point, I know the next boot will work. It didn't work when I wasn't overheating it and applied pressure though, but maybe I have to try a few more times, maybe takes more pressure when not warm.
So this pretty much confirms it's a flex issue of some sort, and likely the BGA like I originally suspected. Some people with this laptop had success with putting a copper shim between the heastsink and GPU, but this laptop proved to be more of a PITA to open than it might be worth, you have to tear down so much...Bottom panel, then screw under the battery, then flip it around and take out the speaker grille, then the keyboard, etc, lots of flipping back and forth just to get at the mainboard
Maybe when I'm less lazy.