for Starlink Cable, Compatiable with Starlink Ethernet Adapter SPX Plug to RJ45 Coupler Adapters Female Connectors Designed for Gen 2 Dishy and Router (Pair for Dish and Router)

Get this, or just the dish end one if you can find it.

You don't need to use the Starlink router at all here. Get a PoE injector in the house, run Ethernet to the dish (and just the dish, no router by the dish), plug your router into the PoE injector's LAN port. It's the same setup I use for my RV, with a 150' Cat6 cable running to the Gen2 dish, and an RJ-45 adapter for the dish's proprietary cable. Here's an adapter you can use. It has adapters for both ends, but you'd only need one end. You just need to make sure the PoE injector you get can put out 56V (rather than 48V) because you'll need the extra voltage for a long run like that and snow melt power draw.