Here's one plan.
Bass, Bogans, the Clippers pick, and the rights to Colton Iverson to Houston for Asik
Use the trade exception to acquire someone. I'd try Paul Millsap, giving Atlanta the 2014 Hawks/Nets pick and the protected Philly pick.
Re-sign Bradley
Use the MLE (not necessarily all of it) to bring in a back-up wing who can play both small forward and shooting guard and who is a good three-point shooter.
The team looks like:
Big man rotation: Asik, Millsap, Sullinger, Olynyk, Anthony, Faverani
Wing rotation: Green, Bradley, free agent wing, Wallace, Chris Johnson
Point guard: Rondo, a veteran minimum salary guy and/or Pressey
Somewhere in the mix: the Celtics 2014 draft pick
I'm not sure if Houston will take that deal. I'm not sure if Millsap will be available. But it's possible that Morey will relent and it's possible that Atlanta will decide to flip Millsap for picks. There might be other guys who fit into those slots. Maybe Thaddeus Young or Anderson Varejao are available over the summer. It's adaptable to whoever is available at certain price points, although Asik and Millsap is probably the ideal combination.
Asik, Millsap, and Rondo would all expire in 2015, meaning you can keep all three or you might have the flexibility to keep two of the three and work out a deal that gives you an upgrade on the third one. You might even keep all three and use draft picks to turn Wallace and/or Green into something better.