= Solution
Factor the extrinsic term as
$$
\boxed{\eta_{22}^{\rm ext}\simeq\eta_{11}\,\frac1{H^2}\,\frac{\beta_2H}{\beta_1+\beta_2H}.}
$$
The first factor is environmental relative <variance>. The second is static susceptibility squared: differentiating the equilibrium relation gives $\delta\log m_2=H^{-1}\delta\log m_1$. The last is the time-averaging factor with effective protein response time $1/(\beta_2H)$. Feedback makes response faster, which by itself weakens averaging, but simultaneously reduces the static gain. These are distinct effects and both must be retained.
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