Handing data from Backbone to CodeIgniter
Check Phil Sturgeon codeigniter restfull controller implementation.
https://github.com/philsturgeon/codeigniter-restserver
It won't be necessary to move to Slim or other framework if you use this rest controller.
I would direct you to http://www.slimframework.com. Nicely implemented REST server using the new PHP closures. Let me know how you get on!