= Solution
For the <harmonic-oscillator recoil matrix element>, using $\psi_1=\sqrt2(x/l_0)\psi_0$ and differentiating the Gaussian transform with respect to $k$,
$$
M_{01}=\frac{\sqrt2g}{l_0}\,i\frac{d}{dk}e^{-k^2l_0^2/4}
=\boxed{-i\frac{gkl_0}{\sqrt2}e^{-k^2l_0^2/4}
=-ig\eta e^{-\eta^2/2}.}
$$
Consequently $|M_{01}|^2/|M_{00}|^2=\eta^2$. Reversing the travelling-wave phase changes the sign of the imaginary amplitude. For small $\eta$, the sideband is weaker by $\eta$ in amplitude and $\eta^2$ in <probability>.
With the printed unnormalized Gaussians, the literal integral would be $-2igkl_0\exp(-k^2l_0^2/2)$. It is distinguished from the physical normalized-oscillator result rather than silently treating inconsistent wavefunctions as normalized.
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