You are right that it's equivalent to compute the degree of the forgetful map. The degree is $3!$, since it's the quotient by $S_3$ permuting the last three markings. If you divide by $S_4$ you get the moduli stack of genus one curves with a distinguished degree $2$ map to $\mathbf P^1$, which is not the same as $\overline M_{1,1}$. 

In fact there should be a canonical isomorphism of stacks $\overline M_{0,4}(B\mathbf Z/2) \cong X(2)$, where $X(2)$ is the modular curve for the full level 2 congruence subgroup, parametrizing elliptic curves with a basis of their 2-torsion. This is easy to show on the interior and I think it will be true along the boundary too, using the modular interpretation of the cusps of $X(2)$ described in Deligne--Rapoport.