= Solution
Let $R=\Delta\tau>0$, $L=\ell_1+\ell_2+\ell_3$ and $D=\prod_{r=1}^3(2\ell_r+1)$. Use exactly the given <constant primordial bispectrum> normalization, without an additional local-template convention factor. Combining it with the large-angle <Sachs-Wolfe effect> <cosmological transfer function> gives the <reduced CMB bispectrum>
$$
b_{\ell_1\ell_2\ell_3}=\left(\frac2\pi\right)^3\frac{4\pi^4f_{\rm NL}\Delta_\zeta^4}{125}\int_0^\infty x^2\,dx\prod_{r=1}^3I_{\ell_r}(x,R),
$$
where $I_\ell(x,R)=\int_0^\infty j_\ell(kR)j_\ell(kx)\,dk$. Each $1/5$ transfer factor has been retained; their product is $1/125$.
Apply the supplied <spherical Bessel product integral> after rescaling $q=kR$:
$$
I_\ell(x,R)=\frac{\pi}{2R(2\ell+1)}\begin{cases}(x/R)^\ell,&0<x<R,\\(x/R)^{-\ell-1},&x>R.\end{cases}
$$
The two branches agree at $x=R$. In the radial integral put $r=x/R$. The factor $R^3$ from $x^2dx$ cancels the $R^{-3}$ from the three kernels. The remaining integral is
$$
\int_0^\infty x^2\prod_r I_{\ell_r}(x,R)\,dx=\frac{\pi^3}{8D}\left[\int_0^1r^{L+2}\,dr+\int_1^\infty r^{-L-1}\,dr\right]=\frac{\pi^3}{8D}\left(\frac1{L+3}+\frac1L\right),
$$
provided $L>0$. This evaluation uses the momentum integrals at fixed radial parameter in the projection, then the radial integral; it does not require an unjustified global exchange of all oscillatory integrals.
The factors $(2/\pi)^3$ and $\pi^3/8$ cancel exactly. Hence the <Sachs-Wolfe projection of a constant bispectrum> is
$$
\boxed{b_{\ell_1\ell_2\ell_3}=\frac{4\pi^4f_{\rm NL}\Delta_\zeta^4}{125(2\ell_1+1)(2\ell_2+1)(2\ell_3+1)}\left[\frac1{\ell_1+\ell_2+\ell_3+3}+\frac1{\ell_1+\ell_2+\ell_3}\right],\qquad \mathcal A=\frac{4\pi^4\Delta_\zeta^4}{125}.}
$$
The all-monopole case $L=0$ has a logarithmically divergent outer radial integral and is not covered by the printed finite formula. Observable CMB analyses remove the monopole and dipole, normally using $\ell_i\geq2$, so this issue is absent there. Angular triangle and parity selection are carried by the triple-<spherical harmonic> geometric factor multiplying the reduced bispectrum.
There is no dependence on the last-scattering distance $R$, and no extra physical scale appears when $\Delta_\zeta^2$ is held constant. At fixed triangle shape and in the range $1\ll\ell_i\ll200$, common rescaling of all multipoles gives
$$
\boxed{b_{\lambda\ell_1,\lambda\ell_2,\lambda\ell_3}\sim\lambda^{-4}b_{\ell_1,\ell_2,\ell_3}.}
$$
This is angular scale invariance in the usual weighted sense; a constant primordial shape does not make the unweighted <reduced CMB bispectrum> independent of angular scale. The exact finite-multipole expression retains the $+1$ and $+3$ terms shown above.
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