I fly the 90-hp no-flaps Super Cub sometimes and have always found it one of the more docile taildraggers. If you are happy in the Pitts, I find it hard to believe that a properly rigged Cub would give you directional trouble and perhaps something is out of whack with it. The brakes are relatively ineffective but you shouldn't need them much for directional control on the roll-out.
I suggest that you ask for the undercarriage geometry to be checked out, as if this happened to me in a Cub I would suspect that the wheel alignment was wrong - perhaps too much toe-in (or insufficient toe out).The bounciness is hard to comment on without knowing something about the state of your strip. On grass I also find the Cub rather bouncy compared to the Pitts - perhaps because it is so light.