Hi Ed,
It's doable, but not easy. You can use a passive PoE injector on both sides to inject and extract power. Injector is bi-directional by its nature.
But... you can't use 12V here. The current at 12V will be extremely high for the PoE injector and Ethernet cable. That's why you need a DC-DC converters to step up the power from the original 12V to 48V, then step it back down to 12V on the Dish side to run the router and 12V mod.