I think the Sox are in a position where it will be pretty hard to upgrade their offense at this point. In fact, it's almost certain that they're bound to suffer considerable regression in terms of scoring runs. The only way that I don't see this happening is if Bogaerts storms out of the gates like Nomar did in 1997. That kind of performance goes a long way to solving a lot of problems. But setting that aside, I think the best course of action now is to "minimize losses". The best way that I can envision this is to maximize platoon splits on their roster.
Top priority would be: No Bradley or Nava against left-handed pitching.
Second priority would be: Protect Carp? He has no career split in the major leagues, but his minor league split is pretty sharp. He also showed a split this year, which makes me think that it might be worth benching him against tough left-handed pitching.
Fortunately, the Sox have Gomes (#27 among qualified players for 3-year WRC+ against LHP), so platooning Nava shouldn't be a problem.
They may be able to use one of their minor league options (Hassan) for Carp protection.
Alternatively, they could pursue
Corey Hart (#7 among qualified players for 3-year WRC+ with a 162 against LHP) for the spot across Carp, but his injury history and potential cost are deterrents. I also don't think it's realistic to expect him to play the short side of a platoon. He'll get better offers elsewhere. Perhaps a better free agency option would be
Jeff Baker, who was released by the Rangers. He doesn't have Hart's numbers against lefties, but he was about as good against them as the reverse-splitting Curtis Granderson over the last three years, with a WRC+ of 124, and can play around the infield.
That leaves Bradley. Because of the way the roster is built, I don't envision the Sox seriously considering someone like Choo or Beltran and forcing Victorino back into centerfield (or, God forbid, reproducing the Choo-in-center experiment). I do think the Sox need a viable defensive option for RF (at least) and preferably one that can play CF, too.
To my eye, there are three right-handed (or right hand favoring switch hitters) options that fill that criteria on the FA market (I'm assuming retirement for Reed Johnson).
Rajai Davis, Franklin Gutierrez, Andres Torres (rehabbing from surgery)
There's been a lot of discussion about Davis already, some about Gutierrez, and none about Torres. Even without considering their respective ages or injury statuses -- i.e. just looking at their splits -- it would appear that Davis is the best option:
3-year splits against LHP (OBP/SLG/WRC+)
Davis (363/455/124+)
Gutierrez (325/476/123+)
Torres (356/358/106+)
When you consider Gutierrez's injury history and Torres' recent surgery, Davis looks like an even better option. But, I wonder if he'll get the opportunity to start somewhere. After all,
Shi Davidi has Davis seeking out "more playing time" than he would get with the Blue Jays. So that leaves the Red Sox with a decision between Gutierrez or Torres. If that's the case, I vote Gutierrez.
Now, if the Sox use one player to fill each of these needs (say, a combination of Gomes, Baker, and Gutierrez), they will be rostering 14 position players. I don't think that will happen. So either they move Gomes and make sure the "Carp protector" is capable of playing 1B/LF (Hart and Baker fit the bill) or they just use the minors as their reservoir for depth. I think they go with the latter option, since so many of those backup options are already in the minors.
So, in the end, this might just be a long way of saying, sign Franklin Gutierrez and call it an offseason.
Or just get really creative...