Probably should be moved here:
SpaceX
To address your question though no that won't work.
That's a basic install video- but the cable from the dish also supplies power, and runs from dish to power supply ethernet at both ends.... then another port on the power supply connects it to the router.
EDIT- in theory you could chain a POE injector and a MOCA adapter at both ends of the coax to connect the starlink router to the dish- removing entirely the starlink power supply and not using it (assuming it doesn't do anything else but supply power and bridge- and assuming you can get POE injectors that supply the required power).
Of course the POE injector and MOCA adapter probably aren't designed to be installed outside and you'd need to supply AC power to it so there's that but maybe you can find industrial ones that are....No idea if Starlink would support such a config- and can't imagine that would be cheaper than just running the one powered ethernet cable... but in THEORY that could possibly work.