Uniqueness of Kantorovich potentials (up to a constant shift) has been analyzed in a very general framework in this work of ours: "On the Uniqueness of Kantorovich Potentials" - https://arxiv.org/pdf/2201.08316.pdf .
Your setting is encompassed in Corollary 2. The key idea is that by optimality, the gradient of the Kantorovich potential is uniquely determined on a subset of $\text{int}(\Omega)$ with full Lebesgue measure. Since for your setting, the Kantorovich potential is locally Lipschitz on $\Omega$ it follows that it is uniquely characterized on the (connected) domain $\Omega$. Let me emphasize that there is no need for $\mu$ or $\nu$ to be absolutely continuous, uniqueness rather depends on the topology of the support of the underlying measures.
Our work also provides sufficient conditions for Kantorovich potentials to be unique if the measures have disconnected support. This could be helpful to extend the scope of your work.