Since I asked the question, I have come to realize that in my application, I know more about $X$ and the action of $G$. For this application, $X$ has finite quotient singularities as it is the coarse moduli space of a smooth DM stack and the action of $G$ comes from an action of the group on this stack in the sense Romagny's paper. Using this, I think I am able to show the desired conclusion. Let me sketch the proof here:
Proposition
Let $\mathcal{M}$ be an orbifold, that is a smooth separated Deligne-Mumford stack with connected coarse moduli space and containing a non-empty open substack which is a scheme. Let the smooth group scheme $G$ act on $\mathcal{M}$ with finite, reduced stabilizers at geometric points. Then the quotient $\mathcal{M}/G$ is again a smooth DM stack.
Proof
Consider the frame bundle $\mathcal{F} = \text{Fr}\left( T_\mathcal{M} \right)$ of the tangent bundle $T_\mathcal{M}$ of $\mathcal{M}$. We know $\mathcal{F}$ is an algebraic space and for $n=\dim(\mathcal{M})$, the group $\text{GL}_n$ acts on $\mathcal{F}$ on the right by
$$ (v_1, \ldots, v_n) . (a_{ij})_{i,j=1}^n = (\sum_{i=1}^n a_{i1} v_i, \ldots, \sum_{i=1}^n a_{in} v_i) $$
and we have that $\mathcal{M} = [\mathcal{F}/\text{GL}_n]$. On the other hand, the action of $G$ on $\mathcal{M}$ induces an action of $G$ on $\mathcal{F}$ by
$$ g . (v_1, \ldots, v_n) = (g_* v_1, \ldots, g_* v_n), $$
where $g_*$ denotes the pushforward under the map $p \mapsto gp$ on $\mathcal{M}$. One shows that both actions are strict actions on the stack $\mathcal{F}$ and that they commute (as pushforward is $\mathbb{C}$-linear) and hence induce an action
$$G \times \text{GL}_n \times \mathcal{F} \to \mathcal{F}$$
with finite reduced stabilizers.
With these preparations done, we simply note that
$$\mathcal{M}/G = [\mathcal{F}/\text{GL}_n]/G = (\mathcal{F}/\text{GL}_n)/G = \mathcal{F}/(\text{GL}_n \times G).$$
The second isomorphism is a consequence of Theorem 4.1 and Remark 2.4 of Romagny's paper. But now $\mathcal{F}$ is a smooth algebraic space and the action of $G \times \text{GL}_n$ has finite, reduced stabilizers at geometric points, so the quotient $\mathcal{F}/(\text{GL}_n \times G)$ is again a smooth Deligne-Mumford stack. qed
Hence instead of first taking the coarse moduli space $X$ of $\mathcal{M}$ and then performing the quotient, one can first take the quotient stack, which is smooth DM, and then take the coarse moduli space, which thus has finite quotient singularities.
I don't know whether this can be used to answer the original question. In Proposition 2.8 of a paper by Vistoli it is shown that any normal complex variety with quotient singularities is the coarse moduli space of a smooth stack. However, to apply this to the question above one would need a way to lift the action of $G$ to a strict action on this stack.
