WKB approximation for a particle on a ring $(E>V)$

You don't use $\psi(s) = \psi(s + n L)$ for $n$ integer, because the ring always has length $L$, its length has nothing to do with $n$. Instead you impose the condition $\psi(s) = \psi(s + L)$, which implies that $$\phi(L) - \phi(0) = 2 \pi n$$ which $n$ is the energy level. Then you get $$n h = \int_0^L \sqrt{2 m (E_n - V(x))} \, dx$$ which is a typical WKB quantization integral, from which you compute the $E_n$ in the usual way.