= Solution
Let $J_+=\lambda$ and $J_-=2\beta m^2$ be the production and loss fluxes, counted in molecules per unit time. At the <steady state>, both equal $J=\lambda$. The restoring <elasticity> and turnover lifetime are
$$
\boxed{H=\frac{d\log J_-}{d\log m}-\frac{d\log J_+}{d\log m}=2,
\qquad\tau=\frac mJ=\frac m\lambda=\frac1{2\beta m}.}
$$
Flux-weighting the magnitudes of the reaction steps, one for production and two for loss, gives
$$
\boxed{\langle r\rangle=\frac{J_+\cdot1+J_-\cdot2}{J_++J_-}=\frac32.}
$$
This is an average over molecular fluxes, as requested, not an average over reaction-event counts; those are different when a reaction removes two molecules at once.
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