To make the question well-posed, I'm going to suppose that we fix $\pi$ and take $q$ to be the smallest integer such that $\pi$ has nonzero invariants under $\Gamma_0(q)$. Then $q$ gives you some information about $\pi$, but I don't think you can reliably expect it to nail down the arithmetic conductor exactly; it is rather a miracle that the paramodular invariants do encode the conductor in all cases, you can't just expect a random family of subgroups always to do this.

The question is, of course, a local one: for $\pi$ a representation of $\operatorname{GSp}_4(\mathbb{Q}_\ell)$ having invariants under the Siegel congruence subgroup $Si(\ell^r) \subseteq \operatorname{GSp}_4(\mathbb{Z}_\ell)$, but not under $Si(\ell^{r-1})$, what can the conductor of $\pi$ be?

For $r = 1$ this can be completely answered using the results of Schmidt's paper "Iwahori-spherical representations" (see the tables in the Roberts–Schmidt book for a handy summary). In particular, if we suppose $\pi$ is generic, then $\pi$ has invariants under $Si(\ell)$, but not $Si(1) = \operatorname{GSp}_4(\mathbb{Z}_\ell)$, if and only if $\pi$ is a representation of type IIa, IIIa or VIa with unramified inducing data; and its conductor is $\ell$ if the type is IIa, but $\ell^2$ if the type is IIIa or VIa. This shows that the dimensions of the Siegel-congruence invariants are definitely not enough to encode the conductor.

(PS. Your question seems to implicitly suggest that the "arithmetic conductors" $N_\pi$ appearing in the functional equation of the degree 4 $L$-factor and the degree 5 $L$-factor are the same. Are you sure about this? I am not sure it is true.)

To make the question well-posed, I'm going to suppose that we fix $\pi$ and take $q$ to be the smallest integer such that $\pi$ has nonzero invariants under $\Gamma_0(q)$. Then $q$ gives you some information about $\pi$, but I don't think you can reliably expect it to nail down the arithmetic conductor exactly; it is rather a miracle that the paramodular invariants do encode the conductor in all cases, you can't just expect a random family of subgroups always to do this.

The question is, of course, a local one: for $\pi$ a representation of $\operatorname{GSp}_4(\mathbb{Q}_\ell)$ having invariants under the Siegel congruence subgroup $Si(\ell^r) \subseteq \operatorname{GSp}_4(\mathbb{Z}_\ell)$, but not under $Si(\ell^{r-1})$, what can the conductor of $\pi$ be?

For $r = 1$ this can be completely answered using the results of Schmidt's paper "Iwahori-spherical representations" (see the tables in the Roberts–Schmidt book for a handy summary). In particular, if we suppose $\pi$ is generic, then $\pi$ has invariants under $Si(\ell)$, but not $Si(1) = \operatorname{GSp}_4(\mathbb{Z}_\ell)$, if and only if $\pi$ is a representation of type IIa, IIIa or VIa with unramified inducing data; and its conductor is $\ell$ if the type is IIa, but $\ell^2$ if the type is IIIa or VIa. This shows that the minimal $r$ at which $\pi$ has nontrivial $Si(\ell^r)$-invariants is not enough to encode the conductor.

(PS. Your question seems to implicitly suggest that the "arithmetic conductors" $N_\pi$ appearing in the functional equation of the degree 4 $L$-factor and the degree 5 $L$-factor are the same. Are you sure about this? I am not sure it is true.)