= Solution
The mass and heat entrainment equations immediately give
$$
\frac{d}{dz}(F-c_pT_0Q)=0.
$$
Use the specified relation $\rho T=\rho_0T_0=\beta$. Then
$$
F-c_pT_0Q=c_pQ(T-T_0)=c_pT_0Q\left(\frac{\rho_0}{\rho}-1\right).
$$
Meanwhile the kinematic <buoyancy flux> is
$$
B=g\frac{\rho_0-\rho}{\rho_0}wb^2
=\frac g{\rho_0}Q\left(\frac{\rho_0}{\rho}-1\right)
=\frac{g}{\rho_0c_pT_0}(F-c_pT_0Q).
$$
Since $T_0$, $\rho_0$ and $c_p$ are constant,
$$
\boxed{dB/dz=0.}
$$
Thus conservation is a consequence of conserved excess heat in this ideal-gas, uniform-ambient model, not merely an assumption that arbitrary non-Boussinesq buoyancy fluxes are conserved in every environment.
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