Assume the wedge does not move ie is anchored to the ground.
For case (i) make the assumption that there is no slipping so no work is done by the frictional force and the decrease in gravitational potential energy is equal to the increase in translational and rotational kinetic energy.
For case (ii) one cannot make the no slipping assumption for all time and so some of the mechanical energy will become heat. This means that the decrease in gravitational potential energy will be greater than the gain in translational and rotational energy.
The reason that there must be slipping at some stage in case (ii) is that when the object reaches $B$ it is not rotating and only has translational motion.
The object cannot instantaneously rotate fast enough for the no slipping condition to be satisfied.
So there must be a period of time during which the object is increasing its rotational motion whilst at the same time slipping.