I am looking for an elementary way to prove the following theorem.

Theorem. Let $\alpha$ and $\beta$ be two simple convex curves in $\mathbb R^2$. Assume $$\mathop{\rm length} \alpha=\mathop{\rm length}\beta$$ and there is a 1-Lipschitz bijecction $f\colon\alpha\to\beta$. Then $f$ is an isometry.

It would be better if the same proof would work for Lobachevsky plane and unit sphere (for the sphere one has to assume that the length of the curves is $<2{\cdot}\pi$).

The proof I know is simple, but it use Alexandrov geometry quite a bit: If we cut from the plane the region bounded by $\alpha$ and glue instead the region bounded by $\beta$ then the obtained space will have curvature $\ge0$ in the sence of Alexandrov and it is easy to show that it has to be isometric to the Euclidean plane. Hence the result.