The sat coaxial on the outside will work for the tailgater , as long as its RG6 , probably is
I`d guess you have a winegard cover with a amp switch on it , small light when its on , the line that goes to the bed room is tied in on the back of it , if you connect that to the coaxial "out to tv" on the sat box you will have standard def tv in the bed room
If you want high def you need a second sat box and another tailgater or a dish that will run two recivers .
Or a HDMI splitter and a long HDMI cable run to the bed rm , with this you will mirror the signal on the main tv ,
If you mirror the signal in either standard or high def you can use
theseto control the sat reciver in another rm