Torsion points on twists of elliptic curves and products of fine modular curves over $\mathcal{M}_{1,1}$ vs over the $j$-line

I think your degree calculation is bogus. You write:

$Y_1(7)^\circ$ is finite etale over $\mathcal{M}_{1,1}^\circ$ of degree $[SL_2(\mathbb{Z}):\Gamma_1(7)] = 48$, in the sense that the fiber category above a geometric point $\text{Spec }k\rightarrow\mathcal{M}_{1,1}^\circ$ is a groupoid with 48 objects, but with isomorphisms between them [...] The resulting scheme theoretic fiber should have degree 24 over $\text{Spec }k$.

But the map $Y_1(7)^\circ \to \mathcal{M}_{1,1}^\circ$ is representable, and this means in particular that the fiber category over $\text{Spec }k$ (or over any scheme for that matter) is an algebraic space, not a stack. So the only isomorphisms in your 48-object category are identities, and the "scheme theoretic fiber" should have degree 48 over $\text{Spec }k$.

I can't say for sure whether or not $A \cong B$, but it's not true that if $E_{1}/K$ and $E_{2}/K$ are elliptic curves with $j(E_{1}) = j(E_{2}) \not\in \{ 0, 1728 \}$ and $P_{1} \in E_{1}(K)$ and $P_{2} \in E_{2}(K)$ both have order $\geq 4$ then necessarily $E_{1} \cong E_{2}$.

In particular, take $E_{1} : y^{2} + xy = x^{3} - 540x + 777615$. This is the "default" elliptic curve with $j = -1/15$, which forces $E_{1}$ to have a cyclic 4-isogeny defined over $\mathbb{Q}$. Let $E_{2}$ be the quadratic twist of $E_{1}$ by $161$ - then $E_{2}$ has a $\mathbb{Q}$-rational 4-torsion point.

Now, let $K/\mathbb{Q}$ be the degree 24 extension obtained by adjoining to $\mathbb{Q}$ the coordinates of a $5$-torsion point of $E_{1}$. Magma tells me that the torsion subgroup of $E_{1}/K$ is isomorphic to $\mathbb{Z}/10\mathbb{Z}$. In particular, $E_{1}$ and $E_{2}$ are not isomorphic over $K$, but $j(E_{1}) = j(E_{2}) = -1/15$ and $E_{1}(K)$ and $E_{2}(K)$ both contain torsion points of order $\geq 4$.