My suggestion is to pick out a cave instructor and talk to him/her. They will tell you how much experience they require before they will take you in their cave class. I would say that if you are comfortable with your CCR, have decent buoyancy and can keep your fins off the floor that you are a candidate for starting your cave training now.
Caving is all about learing how to get out of a cave alive (while respecting the cave), so find an instructor who will challenge you during your training. Then when everything goes south on a subsequent cave dive you will be more prepared to find an exit solution.
I had been cave diving for almost 10 years when I went to a RB. That's a lot of cave time, so I may not represent the type of diver you are seeking. I was CCR certified and didn't take a CCR-cave crossover course; I started in caves with about 40 hours on my RB. I never noticed any difference, other than my profile was a little larger with the RB.
So, just depends on you and your instructor. Good luck.