Its really hard to say what the Pats will do in the draft this year, without having seen what happens in FA.
I think they do select a TE, though. That's one of the team's biggest needs that is unlikely to be addressed via FA.
Will they trade down? I don't know. They traded up for Chandler Jones a couple years ago, they traded down last year and the results from that trade (Collins, Ryan, Boyce -- Blount, arguably) seem pretty good so far. But I don't think BB goes in thinking "I'll trade down". He goes in thinking he'll see how much value he can get with a draft pick, either by trading up with it, by using it, or by trading down with it.