Suppose $M \xrightarrow{p} X$ is a surjective submersion of smooth manifolds. Define the fibered product in this case: $$M \times_X M = \{ (m_1, m_2) \in M \times M | p(m_1) = p(m_2) \}$$
and let $U = \coprod\limits_{\alpha}U_{\alpha}$ such that $\bigcup\limits_{\alpha}U_{\alpha} = M$. Then we have an induced map $U \xrightarrow{\pi}X$ and the following diagram:
$$U \times_X U \times_X U \xrightarrow{\pi_{12}, \pi_{23}, \pi_{13}} U \times_X U \xrightarrow{\pi_{1}, \pi_{2}} U \xrightarrow{\pi} X$$
(please excuse my notation on the multiple arrows, I don't know how to nicely tex that in here.)
**I am looking for a version of Mayer-Vietoris on differential forms induced by this diagram**. Perhaps I can split it into two diagrams (the left three and the right three) and that would still be sufficient. The notes by Moerdijk I am reading through used Bott-Tu as a reference but I couldn't find anything useful in there. If I need to generate my own spectral sequence, that is fine, and I would relish learning opportunity but I wasn't sure if somehow Mayer-Vietoris on these fibered products was already well-known.

Edit: Specifically, given a closed 2-form $\kappa$ on $U \times_X U$, which satisfies $$\pi^*_{12}(\kappa) + \pi^*_{23}(\kappa) = \pi^*_{13}(\kappa)$$ on $U \times_X U \times_X U$, I want to show there exists (via "M-V", according to Moerdijk) a 2-form $\lambda$ on $U$ for which $\kappa = \pi^*_2(\lambda) - \pi^*_1 (\lambda)$. Then, since $d \kappa = 0$, I would have that $\pi^*_2(d \lambda) = \pi^*_1 (d \lambda)$ and so (again by "M-V", according to Moerdijk) I would have $d \lambda = \pi^*(\xi)$ for some closed 3-form $\xi$ on X.

If I was working with fibered products $U \times_M U$ I could think of this all in terms of the Cech-DeRham Complex and I would be comfortable using M-V. But as $U \times_X U$ does not seem quite like "the intersections of the open sets in the cover", I am not quite sure how to approach this.

The context in which I am working is the construction of a 3-curvature via an $S^1$-bundle gerbe, with connection.