Hint: It's a phased-array antenna. It's made of electronics. And those electronics can't be moved away from the antenna elements, because speed-of-light propagation time is critical. And it's wildly-inefficient by design, because by nature it has to burn power transmitting self-cancelling waveforms to steer the RF beam.The best solution is to move the electronics out of the dish to someplace cooler. I doubt the issue is the dish itself getting too hot. Some bonehead engineer put the electronics in the dish which is going to sit in the hot sun regardless of what anyone does. Plus, you're paying $500 for that equipment. Just bad craftsmanship.
You're not going to substantively change the thermal load by juggling whatever cheese-grade ARM application processor they've got shuffling packets to the other end of a wire. It's going to involve actual RF electronics engineers, doing actual RF electronics engineering, while simultaneously trying to streamline manufacturing so that SpaceX isn't losing $500 on every unit they sell.