= Solution
For <proper-time acceleration by reflected laser light>, the spacecraft receives <momentum> $(hf+hf_r)/c=2hf/[c(1+\beta)]$ per incident <photon>. Combining with the impact rate gives
$$
F=\frac{2Rhf}{c}\frac{1-\beta}{1+\beta},\qquad
\frac{d(\gamma\beta)}{dt}=\frac1{\tau_0}\frac{1-\beta}{1+\beta},\qquad
\tau_0=\frac{Mc^2}{2Rhf}.
$$
Because $dt/d\tau=\gamma$ and $d(\gamma\beta)/d\beta=\gamma^3$,
$$
\gamma^2\frac{d\beta}{d\tau}=\frac1{\tau_0}\frac{1-\beta}{1+\beta}
\quad\Longrightarrow\quad
\frac{d\beta}{d\tau}=\frac{(1-\beta)^2}{\tau_0}.
$$
Integrating from rest gives $1/(1-\beta)-1=\tau/\tau_0$, hence
$$
\boxed{\beta(\tau)=\frac{\tau}{\tau+\tau_0}.}
$$
The mirror's <rest mass> is assumed constant and the laser illuminates it continuously; <diffraction> losses, incomplete interception and microscopic recoil are neglected.
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