I re-tried the same procedure last night, but this time without the tinfoil on the GPU. After I'd left it a little while to cool down, I booted it up and the screen lit up straight away. So I figured it probably didn't get enough heat the first time. Left it overnight to settle. Tried again and it booted up perfect first time again! Put it back together and now it's back to a dead screen.

I guess I'm just gonna have to take the heatgun to it after all.