Boston has no way to sign Bosh without a trade with Miami and I don't see the point in Miami taking on players like Bass (even with draft picks). Now if Boston were to add players like Sullinger to the trade, Miami might be more inclined if Bosh was going to leave anyway.
It should be easy for the Celtics to get a team with cap space to accept Bass and Joel Anthony in a salary dump for a pick (or maybe the 2015 Clippers pick and the protected Philly pick).
Assuming no other roster changes, that would leave the Celtics with about $20 million to re-sign Bradley, sign draft picks, and use on free agents. That's if the cap doesn't go up, and I think it is expected to. If the Celtics are willing to cut Pressey's unguaranteed contract and move Faverani, whose contract can be treated as expiring, since its final season is unguaranteed, there might be $24m to split between Bradley, a free agent like Bosh, and a first-round pick. At that point, either spending picks to dump Wallace or renouncing Bradley (if not working out a sign-and-trade, since James and Wade can handle the ball while Bradley plays defense and open jumpers) would work. Trading Olynyk for a future first might also be an option if you want to keep Bradley.
You could then have a roster of Bosh, Sullinger, Green, Rondo, either Wallace or Bradley, Olynyk, a first-round pick, a player or two signed with the MLE, and someone acquired with the Nets trade exception for a draft pick or two. Depending on how it works out, you might not have to renounce Jerryd Bayless and might want to bring him back. Then you fill out the roster with Chris Johnson-type D-League players, veterans who are willing to play for the minimum, and maybe Colton Iverson is NBA-ready.
That may or may not be a contending team, but it could arguably be a player away from contending without that player being an All-Star.