Analyzing the motion of a ball rolling without slipping inside a hemispherical bowl

Your first derivation, using energy, uses two different meanings for the same symbol $\omega$. In one place, you interpret it as

$$\omega = \dot{\theta}$$

the time derivative of the angle of the line from the center of the ball to the center of the bowl with the vertical.

In another place, you interpret $\omega$ as the time derivative of the unnamed angle through which the ball itself has rotated.

These two angles are related to each other by the $r/(R-r)$ factor by which you are off.