You could make your cool tube a closed loop cooling system, then have the rest of box be sealed tight. Intake from your house and exhaust right into your attic ( assuming you have vents in roof/ gable. The majority of the heat from the bulb will never get the chance to heat your box up. With a setup like this the smallest ac you can find would be overkill. Just insulate like you described and you will be left with a 4x4 frig in your attic.