Let $\mathcal{S}$ be the moduli stack (over $\text{Spec }\mathbb{Z}$) of elliptic curves endowed with a trivialisation of its Hodge bundle.

Classically, we know that $\mathcal{S}\otimes \mathbb{Z}[1/6]$ is represented by the affine scheme $$ U=\text{Spec } \mathbb{Z}\left[\frac{1}{6},g_2,g_3,\frac{1}{g_2^3 - 27g_3^2}\right] $$ One can show that it is not representable over $\mathbb{Z}$, so I was wondering what is the coarse moduli space $S$, over $\text{Spec }\mathbb{Z}$, of $\mathcal{S}$?

First of all, can we say something by a priori general reasoning? For instance, do we know a priori that $S$ is a scheme? Affine? etc.

In fact, I have a candidate for $S$: $$ S = \text{Spec }\frac{\mathbb{Z}[c_4,c_6,\Delta^{\pm 1}]}{(c_4^3-c_6^2 - 1728\Delta)} $$ It is a localisation of the ring of modular forms over $\mathbb{Z}$. It is easy to define a natural map $\mathcal{S}\to S$ and I think it is ok to show that $\mathcal{S}(k) \to S(k)$ is a bijection whenever $k$ is an algebraically closed field. But I get confused when trying to prove the universal property.