= Solution
Write $\varphi_G=\varphi_{G,L}+\varphi_{G,s}$. In the <local-type primordial non-Gaussianity> expansion, the term linear in the short field is $\varphi_{G,s}+2f_{\rm NL}\varphi_{G,L}\varphi_{G,s}$. The long-field square contributes to the long background, the mean subtraction is spatially homogeneous, and the short-field square is excluded at the order requested. Hence
$$
\boxed{\varphi_s=(1+2f_{\rm NL}\varphi_{G,L})\varphi_{G,s}}.
$$
This is the long-short coupling measured by a <squeezed bispectrum configuration>.
Because the long potential is constant over the short-scale region, multiplication by $1+2f_{\rm NL}\varphi_{G,L}$ commutes with the linear <cosmological Poisson equation> and any fixed linear smoothing filter. Thus the short <density contrast> satisfies $\delta_s=(1+2f_{\rm NL}\varphi_{G,L})\delta_{G,s}$. Taking the conditional <expected value> over the short <Gaussian random field> gives
$$
\langle\delta_s^2\rangle_s=(1+2f_{\rm NL}\varphi_{G,L})^2\sigma_{G,s}^2,
\qquad
\boxed{\sigma_s^2=\sigma_{G,s}^2(1+4f_{\rm NL}\varphi_{G,L})+O((f_{\rm NL}\varphi_{G,L})^2)}.
$$
The <short-scale variance modulation by local non-Gaussianity> is conditional on the long mode; averaging over that mode as well would erase the term linear in its zero mean.
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