= Solution
At the deterministic feedback equilibrium use $\lambda_2m_1f(m_2)=\beta_2m_2$. The indicated elasticity is
$$
H_{22}=1-\frac{\partial\log f}{\partial\log x_2}=1+h=:H.
$$
The normalized drift becomes $A=\begin{pmatrix}-\beta_1&0\\\beta_2&-\beta_2H\end{pmatrix}$; equal mean birth and death rates still give $D=\operatorname{diag}(2\beta_1/m_1,2\beta_2/m_2)$. Solving the same <covariance> equation yields the <feedback attenuation of gene-expression noise>:
$$
\boxed{\eta_{22}\simeq\frac1{Hm_2}+\frac{\beta_2}{H(\beta_1+\beta_2H)m_1}.}
$$
For $f(x_2)=k/x_2$, $h=1,H=2$, so
$$
\boxed{m_2\simeq\sqrt{\lambda_2km_1/\beta_2},\qquad \eta_{22}\simeq\frac1{2m_2}+\frac{\beta_2}{2(\beta_1+2\beta_2)m_1}.}
$$
The inverse-feedback expression is interpreted near a positive equilibrium; a microscopic birth rate singular at zero needs a separate regularization there.
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