Let me start by writing $1/U$ instead of $U$. The case when $U$ or $1/U$ vanishes would require special handling of course. Then your constraint is $$ \frac{\omega_u}{\omega_v} = \frac{V}{U} \quad \text{or} \quad (U\partial_u - V\partial_v) \omega = 0 . $$ That is, $\omega$ is constant along the flow lines of the corresponding vector field. The vector field $(U\partial_v + V\partial_v)$ is orthogonal to it, with respect to the obvious Lorentzian metric.
Rewriting the sine-Gordon equation using these vector fields gives $$ \frac{1}{4UV} [(U\partial_u+V\partial_v)^2\omega - (U\partial_u-V\partial_v)^2\omega] = \sin \omega . $$ The constraint kills the second term on the lhs, which gives after simple rewriting $$ \frac{(U\partial_u+V\partial_v)^2\omega}{4\sin\omega} = UV . $$ The two vector fields actually commute, so applying $(U\partial_u-V\partial_v)$ kills the lhs, while simplifying its action on the rhs gives $$ \partial_u U = \partial_v V , $$ which together with the underlying hypothesis $U=U(u)$, $V=V(v)$ implies that both functions can only be constants.