= Solution
Choose the emitted beam direction as positive $x$. In the original laboratory frame the sensor moves with <velocity> $-0.2c$; let the laser recede from it with <velocity> $-\beta c$. The collinear <relativistic Doppler shift> from source to laboratory and then to sensor gives
$$
\frac{\nu_{\rm sensor}}{\nu_0}=\sqrt{\frac{1.2}{0.8}}\sqrt{\frac{1-\beta}{1+\beta}}.
$$
At the detection threshold this ratio is $633/1000$. Hence
$$
q=\frac{(633/1000)^2}{1.5}=0.267126,\qquad
\boxed{\beta=\frac{1-q}{1+q}\simeq0.578375.}
$$
The minimum collinear speed is therefore \b[about $0.578c$, away from the approaching sensor], while the laser is still aimed towards it. The source must recede fast enough to overcome the sensor's approach-induced blueshift. A larger recession speed gives a longer received <wavelength>, if the detector has no additional upper cutoff.
As a check, the required relative recession speed is $(1-0.633^2)c/(1+0.633^2)$; <relativistic velocity addition> gives this same value from $(\beta-0.2)c/(1-0.2\beta)$.
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