= Solution
Write $A_k=H/\sqrt{4\epsilon M_{\mathrm{Pl}}^2k^3}$, so $u_k(0)=A_k$. For the cyclic term with the first field undifferentiated, insertion into the <in-in formalism> gives
$$
a^2\prod_ru_{k_r}(0)\,u_{k_1}^*u_{k_2}^{*\prime}u_{k_3}^{*\prime}=\frac{H^4}{64\epsilon^3M_{\mathrm{Pl}}^6(k_1k_2k_3)^3}\,k_2^2k_3^2(1-ik_1\tau)e^{iK\tau},
$$
where $K=k_1+k_2+k_3>0$. The two differentiated modes contribute $\tau^2$, cancelling the $\tau^{-2}$ in $a^2$.
Along the vacuum-selected contour, the lower boundary vanishes. The elementary integrals are
$$
\int_{-\infty(1-i\varepsilon)}^0e^{iK\tau}\,d\tau=-\frac{i}{K},\qquad \int_{-\infty(1-i\varepsilon)}^0\tau e^{iK\tau}\,d\tau=\frac{1}{K^2},
$$
and therefore
$$
\int_{-\infty(1-i\varepsilon)}^0(1-ik_1\tau)e^{iK\tau}\,d\tau=-i\left(\frac1K+\frac{k_1}{K^2}\right).
$$
Multiplying by $4iM_{\mathrm{Pl}}^2\epsilon^2$, taking the real part and summing the three cyclic choices gives the <curvature bispectrum from a zeta zeta-prime-squared interaction>:
$$
\boxed{\begin{aligned}
\langle\zeta_1\zeta_2\zeta_3\rangle_c&=(2\pi)^3\delta^{(3)}(\mathbf k_1+\mathbf k_2+\mathbf k_3)B^{\mathrm{sf}},\\
B^{\mathrm{sf}}&=\frac{H^4}{16\epsilon M_{\mathrm{Pl}}^4(k_1k_2k_3)^3}\sum_{\mathrm{cyc}}k_2^2k_3^2\left(\frac1K+\frac{k_1}{K^2}\right).
\end{aligned}}
$$
This matches the final normalization. The coefficient requires all six connected <Wick contractions>.
The calculation assumes the Gaussian <Bunch-Davies vacuum>, tree order in the specified cubic interaction, effectively constant $H$ and $\epsilon$ during the integral, the approximate de Sitter <scale factor>, and observation after the modes freeze as $\tau\to0^-$. The contour tilt is kept until the early boundary has been eliminated; an undamped real-axis integral cannot simply discard that boundary. The <reduced Planck mass> convention is the one in the supplied <De Sitter curvature mode function>. Other cubic interactions and nonlinear field redefinitions are outside the specified model, so this is not by itself the complete bispectrum of a general single-field action.
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