I find that cabs are readily available at all Disney resorts. You can call bell services at the CBR and ask them to have the cab pick you up at a location convienent to your room, or at Old Port Royale.
I'm not sure how much a cab will cost from the CBR to the parks, as I've never taken one to the park from the CBR, but my guess based on taking other cabs is around $10 to DHS, $15 to Epcot, and $20 to the MK (the TTC is where you'd be dropped off). I don't know if those fares are accurate, but I think it should be witin a few dollars.