Let $\mathcal{M}$ be a smooth 1-dimensional Deligne-Mumford stack with finite diagonal, and let $M$ be its coarse moduli scheme (all over some field $k$). Suppose $M$ is also smooth over $k$, and that all local deformation problems for $\mathcal{M}$ are pro-representable (I think this is already implied by the smooth + DM conditions?)

Let $x_0 : \text{Spec }\bar{k}\rightarrow\mathcal{M}$ a geometric point, and let $\bar{x_0}\in M$ be its image in $M$.

Let $\mathcal{R}$ be the completion of the etale local ring at $x_0$ (colimit of global sections of scheme etale neighborhoods of $x_0$), and let $R$ be the completion of the etale local ring of $M$ at $\bar{x_0}$.

This yields a natural map $R\rightarrow\mathcal{R}$ (both are isomorphic to a 1-variable power series ring over $\bar{k}$), whence a map $$p : \text{Spec }\mathcal{R}\rightarrow\text{Spec }R$$

I believe the infinitesimal lifting property of etale morphisms then implies that $\mathcal{R}$ is also the universal deformation ring of the object corresponding to $x_0$ (am I missing any conditions here?).

Let $X_0/\bar{k}$ be the object corresponding to $x_0$, and let $X^\text{univ}/\mathcal{R}$ be the universal deformation of $X_0$ over $\mathcal{R}$.

I would like to say something like: $p$ is finite flat totally ramified with ramification index equal the order of the group of automorphisms of $X_0$ modulo the subgroup of those automorphisms which extend to an automorphism of $X^\text{univ}/\mathcal{R}$.

Is this true? What is a good reference? If it's false, what is the relation between the universal deformation ring of $x_0$ and the complete etale local ring of $x_0$?

Actually, some calculations of mine seem to indicate that may be false, though it's possible I've made a mistake somewhere else...