= Solution
Evaluate all derivatives at zero long perturbation, where $\nu=\delta_c/\sigma_{G,s}$, and write $\bar n(\nu)$ for the unperturbed abundance. The <peak-background split> changes the <halo peak height> to
$$
\nu_{\rm loc}=\frac{\delta_c-\delta_L}{\sigma_{G,s}(1+2f_{\rm NL}\varphi_{G,L})}
=\nu-\frac{\nu}{\delta_c}\delta_L-2f_{\rm NL}\nu\varphi_{G,L}+\cdots.
$$
The <chain rule> in the specified <bias expansion> therefore gives
$$
\boxed{b_{10}=-\frac{\nu}{\delta_c}\frac{d\log\bar n}{d\nu},\qquad
b_{01}=-2f_{\rm NL}\nu\frac{d\log\bar n}{d\nu}
=2f_{\rm NL}\delta_c b_{10}}.
$$
No explicit abundance law is specified, so these derivatives are the general answer. For example, if $\bar n$ has the <Press-Schechter halo mass function> dependence $\nu e^{-\nu^2/2}$, then $b_{10}=(\nu^2-1)/\delta_c$ and $b_{01}=2f_{\rm NL}(\nu^2-1)$.
For the linear long mode, the <cosmological Poisson equation> gives $\varphi_{G,L}(\mathbf k)=\delta_{G,L}(\mathbf k)/\alpha(k)$. Consequently the deterministic <galaxy bias> and <power spectrum> are
$$
\delta_g(\mathbf k)=\left[b_{10}+\frac{b_{01}}{\alpha(k)}\right]\delta_{G,L}(\mathbf k),
\qquad
\boxed{P_g(k)=b_{10}^2\left[1+\frac{2f_{\rm NL}\delta_c}{\alpha(k)}\right]^2P_m(k)}.
$$
Here $P_m$ is the linear <cosmological density power spectrum>; stochastic tracer noise has been omitted. This <scale-dependent halo bias from local non-Gaussianity> is proportional to $k^{-2}$ in the specified $\alpha(k)=2k^2/(3H^2\Omega_m)$ convention.
For $b_{10}\ne0$, plot the ratio $P_g/(b_{10}^2P_m)$ to separate the effect from the shape of $P_m$. Positive $f_{\rm NL}$ increases the ratio as $k$ decreases. Negative $f_{\rm NL}$ first suppresses it, gives a zero when $\alpha(k)=2|f_{\rm NL}|\delta_c$, and then makes it rise again after the effective <galaxy bias> changes sign. The auto-<power spectrum> never becomes negative. A truncation $P_g\simeq b_{10}^2[1+4f_{\rm NL}\delta_c/\alpha(k)]P_m$ is valid only when $|2f_{\rm NL}\delta_c/\alpha(k)|\ll1$ and must not be extrapolated through the zero. Within the deterministic expression, both signs give $P_g\propto k^{-4}P_m$ at sufficiently small $k$. For a nearly scale-invariant primordial spectrum, $P_m\propto k^{n_s}$ in the large-scale transfer limit, so this formal asymptote is $k^{n_s-4}$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-312-bias-sketch.png]
{title=Scale-dependent galaxy bias}
{description=Schematic deterministic power ratio for equal positive and negative coupling magnitudes, with $\alpha\propto k^2$. The negative-coupling cancellation scale defines $k_{\rm cancel}$; the vertical axis is logarithmic away from a small linear region around zero.}
There is a convention issue in calling the specified abundance response an observed galaxy overdensity. The threshold response is naturally a <linear Lagrangian halo bias>. If $b_{10}$ instead denotes the physical <linear Eulerian halo bias>, the <Lagrangian-to-Eulerian linear bias relation> adds one to the density coefficient: $b_{01}=2f_{\rm NL}\delta_c(b_{10}-1)$, and $P_g=[b_{10}+b_{01}/\alpha]^2P_m$. The boxed formulas follow the abundance expansion explicitly supplied in the paper. The Newtonian treatment also cannot be extrapolated beyond its physical large-scale domain without relativistic projection effects.
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