I think he stays.
After thinking on it long and hard draft picks are over valued. And based on how the Pats have traded picks for proven players this off-season it appears that BB agrees.
Consider the last 2 1st, and last 2 2nd round picks the Patriots have made. Malcolm Brown and Dominique Easley. Jordan Richards and Cyrus Jones. Are any of those impact players?
Even if the Brown's gave up #1 overall and we picked Garrett. The thing is we would have to pay him ~7m/year for his 1st 4 years. So he pretty much needs to immediately be an average level starter for him to be fair value by himself. And that excludes the opportunity cost of losing JG. So really Garrett would have to immediately be an impact player for the deal to be good for the Pats.
As a comparison instead of paying JG + ~$7m year for Garret, BB paid $13m a year for Gilmore, who based on his contract we can pretty safely assume BB expects to be an immediate impact player.
So at this point I think BB sees more value in trying to finding starters in mid-round+ draft picks(think Logan Ryan, Duron Harmon, Flowers, Joe Thuney, Malcolm Butler,etc) and then adding proven NFL talent such as Gilmore and Cooks through FA, trading draft picks. And honestly I find it hard to disagree with that approach.