For SHM we require that the restoring force (or torque) should be directed toward the equilibrium position and be proportional to the displacement $\theta$.
The height of the centre O of the ring, taking its lowest point at $\theta=0$ as origin, is
$h(\theta)=(R-r)(1-\cos\theta)$.
When the centre O of the ring moves anticlockwise through angle $\theta$ from the vertical, the point of contact P between the ring and cylinder moves a distance $s=R\theta$ to the right. Relative to the line COP joining the centre of the cylinder C to point of contact P, the ring turns clockwise through an angle $\phi=s/r=(R/r)\theta$. Relative to the vertical the ring turns clockwise through angle
$\phi-\theta=(R/r-1)\theta=k\theta$
So the height of the bead above the origin is
$H(\theta)=h+r\cos(k\theta)$
The ring and bead each have mass $m$, so the potential energy of the system (ring plus bead) is
$V(\theta)= mgh+mgH=mg[2(R-r)(1-\cos\theta)+r\cos(k\theta)]$.
Without loss of generality we can take $mg=1$. Then the restoring torque is
$\tau=-\frac{dV}{d\theta}= -[2(R-r)\sin\theta-kr\sin(k\theta)] \approx -[2(R-r)-rk^2]\theta$
for small angles $\theta$.
This shows that, for small angles, the restoring force $\tau$ is approximately proportional to displacement $\theta$, as required. We also require that $\tau$ should be -ve when $\theta$ is +ve, ie that potential energy should increase away from the equilibrium position : $\frac{dV}{d\theta}\gt 0$.
Using $R/r=\rho$ we require :
$2(\rho - 1) - k^2 = 2\rho -2 - (\rho^2-2\rho+1) > 0$
$0 > 3 - 4\rho + \rho^2 =(1-\rho)(3-\rho)$
ie
$1 < \rho=R/r < 3$
The least ratio for SHM is $r/R=1/3$. (Note that physically we cannot have $r/R>1$ anyway.)
Note : It might also be possible for the ring to perform SHM if it starts in other positions, ie not at the lowest point of the cylinder where $\theta=0$. See Equilibrium and movement of a cylinder with asymmetric mass centre on an inclined plane for an example in which an object oscillates on an inclined plane. (However, I did not prove that those oscillations are SHM.)