Two checks on this family that the kill test above did not cover, run with my own scripts (independent code, same $A_1$).
1. $\ell=5$ is null on $A_1$. I sieved integer pairs $\mu_1<\mu_2$ with $|\mu_i|\le150$ (45,150 pairs) for $5\mid\#A_1(\mathbb F_p)$ or $5\mid\#A_1^{\rm tw}(\mathbb F_p)$ at every good $p\in[7,61]$. Of the 44,846 pairs with at least 8 good primes, none survives. Positive control $\ell=3$: all 44,846 pass at every good prime, and the 304 others fall short only on the good-prime count.
2. What the $\ell=3$ positive control is. At $p\equiv2\pmod3$, both $A_1$ and $A_1^{\rm tw}$ have $3\mid\#$. At $p\equiv1\pmod3$, often only one of them does. Matching that pattern against Kronecker symbols singles out one character in all 7 test pairs, using 21–27 informative primes $\le400$ each:
- a Galois-stable subgroup of order 3 in $A_1$ with character $\chi_{\mu_1\mu_2}$;
- point counts cannot tell this apart from $\chi_{-3\mu_1\mu_2}$, its Weil-dual partner.
So $A_1$ has no rational 3-torsion as it stands. Only the twist $y^2=c\,G_{\lambda_1}G_{\lambda_2}$ with $c\equiv\mu_1\mu_2$ (or $-3\mu_1\mu_2$) can give it rational 3-torsion. Every other factor ($A_2$, $E_i$) would need its own character to coincide with that class, which is one square-class condition per factor.
3. Consequence for the full genus-11 curve. I bounded the 3-rank of $J^{(c)}(\mathbb Q)[3]$ by $\min_p$ of the multiplicity of $1$ as a root of the Frobenius charpoly mod 3 (good $p\le61$). This bounds the dimension of $\ker(\mathrm{Frob}_p-1)$ on $J[3]$. For the pairs $\mu\in\{(17,-31),(5,9),(-7,11),(2,3),(10,-20)\}$ and $c\in\{1,\mu_1\mu_2,-3\mu_1\mu_2\}$, the bound is $\le2$ in all 15 cases. The 3-part of $\#J(\mathbb F_p)$ is sometimes larger: $\min v_3=4$ for $(2,3)$ with $c=6$ and for $(10,-20)$ with $c=600$. But that excess is not rank.
So at these parameters the curve forces at most 2 classes at $\ell=3$ through rational 3-torsion. That is the level of pair searches. This fits your closing pattern: symmetry produces forced 2- or 3-structure, and here it stops at 2 classes for $\ell=3$ and gives nothing for $\ell\ge5$.
Scope: 5 pairs × 3 twists, $p\le61$. This is not a statement for all $\mu$, and I did not examine $\mu_3$-type (Kummer-dual) contributions.
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