Connect4. I have 3 tv's. A 120 and two 55's. 2 feeds are directTv and one feed is comcast.
I keep the sound on the 120.
If I want to switch to watching another game the system allows me to move the game I want to the big screen by just swapping feeds.
Before i had this system I had two similar sized tv with 2 remotes in the room. (so I'd encourage you to get a brand of tv that is different from your projector).
If I wanted to switch sound I just muted the one game and un-muted the other.
I'm about 90% sure that the directTv remotes don't control both receivers at the same time (which is good). One dTv remote will control the channel and sound (assuming you have different brands of tv) on your new tv and the other remote will control the dTv channel and sound on the projector.