Hdmi over coax and use your existing outside coax with a coax to the receiver module then hdmi to the tv .. there is a transmitter you connect to the hdmi out on the sat box then the coax to the outside. You’ll need a hdmi splitter if you want to watch the inside tv also 100 bucks up for the both the transmitter & receiver and another 30-50 for the hdmi splitter.
The cheap way , and this is how I currently send my wally to the bedroom tv is with a component ( the yellow white red out ) to coax adapter commonly called a game adapter. It sends the signal std definition to the rear the on coax on Channel 3 or 4. Turn the tv on to Channel 3 and change Channel’s with the sat tuner. The picture isn’t high def but it’s passable. All in 12 bucks.