I want to start by considering a familiar congruence subgroup of the integral symplectic group $\text{Sp}_{4}(\mathbb{Z})$. For a positive integer $N$, let $\Gamma _{0}^{(2)}(N) \subset \text{Sp} _{4}(\mathbb{Z})$ be the obvious analog of the congruence subgroup $\Gamma _{0}(N) \subset \text{SL} _{2}(\mathbb{Z})$. That is, writing your symplectic matrix in the usual $2 \times 2$ block form, the lower left entry should be $0$ modulo $N$. (I would write out the matrix for clarity, but there are some frustrating and inexplicable issues with the Latex compiler on this site right now...)
Anyways, given a principally-polarized Abelian surface $A$, the Weil pairing induces a symplectic structure on the $N$-torsion points $A[n]$. If I'm not mistaken, this remains true if $A$ is $(1,d)$-polarized, with $(d,N) = 1$. On page 5 here, they explain that the moduli space $\mathbb{H}_{2}/\Gamma^{(2)} _{0}(N)$ parameterizes pairs $(A,G)$ consisting of a principally-polarized Abelian surface $A$ together with an isotropic $2$-dimensional subspace $G \subset A[N]$. They at least show this for $N=p$ prime, but I imagine it holds for any $N$? Though you must of course work with symplectic modules instead of vector spaces.
My main question is the following. What if I am interested $(1,N)$-polarized Abelian surfaces together with $N$-torsion data? There is a well-known paramodular group $K(N) \subset \text{Sp} _{4}(\mathbb{Q})$ such that $\mathbb{H} _{2}/K(N)$ parameterizes $(1,N)$-polarized Abelian surfaces. However, I believe that in this case $A[N]$ fails to carry symplectic structure. So if one were to introduce the obvious analogous congruence subgroup $K _{0}(N) \subset K(N)$, the extra torsion data cannot be interpreted as counting isotropic subspaces. Can the original problem be reinterpreted so that it extends to this case? In other words, can those isotropic subspaces be reinterpreted as, for example, the kernel of certain isogenies? And does this perhaps lend to the correct modular interpretation of $\mathbb{H} _{2}/K _{0}(N)$?
