But the only way you'd get denser air into the airbox would be to cool down all the ambient air.
Peltier device on the intercooler would be best, I think it would be better on the outlet side. Difficult to mount with the stock intercooler setup (they have plastic end tanks), unless you can put one on the crossover pipe.
End tank of a forge/whatever FMIC would be a good place.
Might be good on the inlet manifold itself, but the heat output would add to the engine bay heat
Bill ran peltier (a) device(s) for a while, have you done a search?
Edit: I wonder why your airbox is getting hot, I can't say I've noticed mine being hot
It might get hot after it's been stood for a bit after a run, but I don't think it gets hot during a run...
Also, have search on audiworld - I seem to remember Steve Schwing doing some testing on airbox temps and the effect on IAT.