Throught this answer, I'll be referencing Kubert--Lang, "Modular Units" Ch 2, sections 1 and 2, and Ch 3 section 4.
Each of your functions is a ratio of the Siegel functions of the form $$A_N(\tau)=\left(\frac{g_{\frac1N,0}}{g_{\frac aN,0}}(N\tau)\right)^m.$$
If $r=(r_1,r_2)\in \mathbb Q^2$, $B_2(x)=x^2-x+1/6$ is the second Bernoulli polynomial, $q=e^{2\pi i \tau}$ as usual, and $\zeta=e^{2\pi i r_2}$, then the Siegel function $g_a$ is defined by $$ g_a(\tau)= -q^{\frac12 B_2(r_1)}e^{2\pi i \frac{r_2(r_1-1)}2}\prod_{n=0}^\infty(1-q^{n+r_1}\zeta)(1-q^{n+1-r_1}\zeta^{-1}). $$ The Siegel functions $g_a$ are modular units, meaning they're modular functions whose zeros and poles are supported at cusps.
If $\gamma\in \text{SL}_2(\mathbb Z)$, then $$ g_a|_0\gamma=g_{a\gamma}. $$ Moreover, we can reduce $r_1$ and $r_2$ modulo $\mathbb Z$, or change signs of both $r_1$ and $r_2$, if we introduce a root of unity. However, the $m$'s in your ratios satisfy $m(1-a^2)\equiv 0 \pmod N$ if $N$ is odd and $\pmod{2N}$ if $N$, and so using Theorem 3.4.1 of Kubert--Lang we find we can ignore this extra factor so the transformations rules becomes this simple. Note, the theorem is written in terms of Klein functions $t_a=g_a\Delta^{-\frac1{12}}$. Using these transformation rules, it's easy to see that all of your functions are invariant under $\Gamma_1(N)/\{\pm1\}$.
The specific groups you have considered are ones where this group is genus $0$, and your functions $A_N$ are Hauptmodln with a unique zero at the cusp $\infty$, a constant $C_0$ at the cusp $0$, and a unique pole at one other cusp. We can calculate $C_0$ using the formulas above: If $\zeta_{2N}=e^{2\pi i \frac{1}{2N}}$, then the constant is given by $\left(\frac{\zeta_{2N}-\zeta_{2N}^{-1}}{\zeta_{2N}^{a}-\zeta_{2N}^{-a}}\right)^m$.
Since $A_N$ is a hauptmodl, its image under the Fricke involution is a rational function in $A_N$. Since it now has a unique $0$ at the cusp $0$ (Where $A_N$ has the constant $C_0$) and the constant $C_0$ at the cusp infinity (Where $A_N$ has a zero), we must have $$ A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}=\frac{-A_N+C_0}{X\cdot A_N+1}, $$ for some $X$, determined by the movement of the pole. If the pole does not move, then $X=0$. Otherwise, $-1/X$ is the constant of $A_N$ at the cusp where $A_N|_0\begin{pmatrix}0&-1\\N&0\end{pmatrix}$ has its pole.
All in all, this suggest you might have hope for $N=7,9$ and $10$, which all give genus $0$ groups. Unfortunately The ratios of the type you've described above aren't hauptmodln. You might be able to find similar results if you allow for larger order products. For instance, if $P\equiv 1\pmod 4$ is prime and $\chi$ is the quadratic character mod $P$, then the product
$$
B_P(\tau):=\prod_{a=1}^{\frac{P-1}2} g_{\frac aP,0}(p\tau)^{\chi(a)}
$$
should satisfy $$\left(B_P\right)^m|_0\begin{pmatrix}0&-1\\P&0\end{pmatrix}=C_0-\left(B_P\right)^m$$ for appropriate $m$, since this is modular on a group with three distinct cusp: The cusp $\infty$ where the function has all its zeros, the cusp $0$, where the function has a constant, and one other cusp fixed by the Fricke involution, where the expansion is given by
$$\left(B_P\right)^m|_0\begin{pmatrix}a&b\\c&d\end{pmatrix}=\pm \left(B_P\right)^{-m},$$
where $c\equiv 0\pmod P$ and $\chi(d)=-1$.