It's very important to define things carefully the problem of interest before constructing the moduli space / stack. In the most general setting you want to carefully define the moduli functor, but you should always at least have a precise idea of what the field-valued points are.

For instance, you don't specify how many branch points / what genus you want the covers to have. I guess you want two branch points, and genus zero.

You also have to define the stability condition for a cover. The most reasonable choice is probably Kontsevich's definition of stable maps, where double cover is defined to be any stable map of degree 2, but you could also take a double cover to literally be a two-to-one map with no constant components. This is no problem in the genus zero case because there are no stable covers with constant component.

Your construction is completely valid for the coarse moduli space, although of course you are right that you have to quotient by $\mathbb Z/2$ switching the two marked points. However, taking that quotient in stacks gives the wrong answer, because it implies you have extra automorphism when the two points coincide. Assuming you interpret this as a (stable) nodal curve, that extra automorphism is meaningless. Instead it's better to use the isomorphism $(\mathbb P^1 \times \mathbb P^1 ) / (\mathbb Z/2) = \mathbb P^2$, which you can see explicitly by viewing a degree $2$ branch divisor as a line inside the space of sections of $\mathcal O(2)$.

To actually construct the moduli stack in this setting, you have to take the universal family of double covers over $\mathbb P^2$ and quotient out by the hyperelliptic involution. This will give the extra automorphism when the marked point coincides with a branch point. This works because in general, the moduli stack of varieties with one marked point is the universal family over the moduli stack, and because the moduli stack here is $\mathbb P^2 / (\mathbb Z/2)$ where the $\mathbb Z/2$ action we mod out by is trivial.

With greater numbers of branch points, this won't quite work, as when three or more branch points coincide the double cover is no longer stable, and you will need to do some blow-up, or other tricks, to deal with it.

It's very important to define things carefully the problem of interest before constructing the moduli space / stack. In the most general setting you want to carefully define the moduli functor, but you should always at least have a precise idea of what the field-valued points are.

For instance, you don't specify how many branch points / what genus you want the covers to have. I guess you want two branch points, and genus zero.

You also have to define the stability condition for a cover. The most reasonable choice is probably Kontsevich's definition of stable maps, where double cover is defined to be any stable map of degree 2, but you could also take a double cover to literally be a two-to-one map with no constant components. This is no problem in the genus zero case because there are no stable covers with constant component.

Your construction is completely valid for the coarse moduli space, although of course you are right that you have to quotient by $\mathbb Z/2$ switching the two marked points. However, taking that quotient in stacks gives the wrong answer, because it implies you have extra automorphism when the two points coincide. Assuming you interpret this as a (stable) nodal curve, that extra automorphism is meaningless. Instead it's better to use the isomorphism $(\mathbb P^1 \times \mathbb P^1 ) / (\mathbb Z/2) = \mathbb P^2$, which you can see explicitly by viewing a degree $2$ branch divisor as a line inside the space of sections of $\mathcal O(2)$.

The moduli stack of double covers of $\mathbb P^1$ is not quite $\mathbb P^2$, because we have to account for the hyerelliptic involution. In fact, it is the quotient of $\mathbb A^3 - \{0\}$ by $\mathbb G_m$, where $\mathbb G_m$ acts via scalar multiplication by the square. To see this, represent a double cover as the relative Spec of $\mathcal O_{\mathbb P^1} + \mathcal O_{\mathbb P^1}(-1)$, with multiplication given by $(a_1,b_1)\times(a_2,b_2) \to (a_1a_2 + f b_1b_2, a_1b_2 + a_2b_1)$ for $f$ a nonzero section of $\mathcal O_{\mathbb P^1}(2)$. This defines a double cover, every double cover can be defined this way, and equivalences between the double covers defined by $f_1$ and $f_2$ are parameterized by $\lambda \in \mathbb G_m$ with $f_2=\lambda^2 f_1$. Locally, this is equal to the quotient of $\mathbb P^2$ by $ \pm 1 \subset \mathbb G_m$, but not globally (as I earlier claimed).

To construct the moduli stack in this setting, you have to take the universal family of double covers over this stack. In other words, you take the universal family of double covers over $\mathbb A^3 - \{0\}$, constructed by exactly this relative Spec, and quotient by this $\mathbb G_m$ action. This works because in general, the moduli stack of varieties with one marked point is the universal family over the moduli stack of varieties. For curves, this is true with Deligne-Mumford or Kontsevich stability conditions even if the marked point touches a singularity, although this case requires blowing up the universal family of curves over the moduli space of curves with marked points to produce the universal family of curves with marked points.

With greater numbers of branch points, this won't quite work, as when three or more branch points coincide the double cover is no longer stable, and you will need to do some blow-up, or other tricks, to deal with it.