Couple of points. You must make sure that the BV1 has the link module fitted or it won't work. The LS model is the one to get. The Avant does not have to be on to access the channels.
As I said in the other thread, the Tivo does not seem to be covered in the STB list available here. Therefore working it via the B&O system would seem unlikely.
There are video senders - most seem to be wireless - they don't work in my house! Walls too thick!