Well, it's good to know it's not the wiring. And it's a good thing that chip shuts off before it sets your guitar on fire. I have a feeling a heat sink is going to have to be kind of large in order to function -- that's the point, right? I'm thinking it's a big chunk of metal used to dissipate heat.
I'm sure others have made this suggestion, but I'll encourage you again to just make a separate enclosure for the amp, and get it out of the guitar. Firstly, the guitar will likely sound better when it's not stuffed full of all this stuff. Secondly, I don't think it's a good idea to put anything that generates substantial heat inside a wooden guitar that you value. Also, it occurs to me that sitting inside this tight wooden box, you're not getting a lot of ventilation to this thing, so it's going to be quicker to heat up. So far you haven't made any irreversible changes to the guitar, and I'd like to see you continue that. In a separate box, you could put as many holes/vents in the outside as you like and get this thing cooled off.
I've seen these done in fun ways, too. Years and years ago, a buddy of mine made a belt-mountable amp out of a He-man shower radio. I'm sure you could find an interesting box to fit this stuff into.