I'd trade Carp, Workman or Brentz for Ryan Hanigan. He's an elite defensive catcher, who's lead the league in caught stealing percentage in each of the last two years and is one of the best catchers in baseball at handling a pitching staff and framing pitches.
I'd give Hart a one year pillow contract. Fenway would be the perfect place for him to rebuild his value and it between Napoli walking and Hart walking after next year, it could potentially net us two draft picks.
I'd bid $71 million for Mashairo Tanaka and then sign him to a $75 million deal. He wouldn't cost a draft pick and the $12.5 million cap hit would be very reasonable. He's not Yu Darvish, but he does project to be like Hiroki Kuroda.
If that isn't enough to get Tanaka, I'd be willing to go five years on a contract for Shin-Soo Choo. He's on base percentage would be great at the top of the lineup, and we have a guy in Gomes who would mitigate his issues against lefthanded pitchers.
I'd also see if we could get anything useful for Dempster, and Nava or Carp, Mookie Wilson, Bryce Brentz, etc. If we could get a backup option in centerfield like Lorenzo Cain (poor man's Peter Bourjos), that would be ideal. I believe he has minor league options left as well. We may also be able to upgrade the bench and get someone better than Brock Holt as our backup shortstop/third baseman, or add another quality reliever.