= Solution
Let $G=\Phi_G$ and $f=f_{NL}^{\rm local}$. The <Gaussian random field> <covariance> in Fourier space is
$$
\langle G(\mathbf k)G(\mathbf p)\rangle=(2\pi)^3\delta^3(\mathbf k+\mathbf p)P(k).
$$
The <Fourier transform> of the centered quadratic term is
$$
Q(\mathbf k)=\int\frac{d^3q}{(2\pi)^3}G(\mathbf q)G(\mathbf k-\mathbf q)-(2\pi)^3\delta^3(\mathbf k)\langle G^2\rangle.
$$
The Gaussian three-point function vanishes. At first order in $f$, the only terms are $f\langle Q_1G_2G_3\rangle$ and its two cyclic placements. For the first, <Wick theorem> pairs four Gaussian factors in three ways. Pairing the two factors inside $Q_1$ together is canceled by the mean subtraction. Pairing one of those factors with $G_2$ and the other with $G_3$ gives $P(k_2)P(k_3)$ and the overall momentum delta; exchanging the two factors gives the same contribution. Thus
$$
\langle Q_1G_2G_3\rangle=2(2\pi)^3\delta^3(\mathbf k_1+\mathbf k_2+\mathbf k_3)P(k_2)P(k_3).
$$
Adding the three placements gives
$$
\boxed{\langle\Phi_1\Phi_2\Phi_3\rangle_c=2f_{NL}^{\rm local}(2\pi)^3\delta^3(\mathbf k_1+\mathbf k_2+\mathbf k_3)[P(k_1)P(k_2)+P(k_2)P(k_3)+P(k_3)P(k_1)]+O(f^3).}
$$
There is no second-order term because it would contain five centered Gaussian factors. The two-point function receives its first correction at $f^2$, whereas the displayed <local-type primordial non-Gaussianity> gives a three-point correlation already at $f$. For $P\propto k^{-3}$ the squeezed limit is $B_\Phi(q,k,k)\simeq4fP(q)P(k)$; this explains the strong correlation between a long potential and short-scale power. A finite <variance> or regulator is needed to define the subtracted $\langle G^2\rangle$, as noted in part (b).
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