 Solution

A spatial rotation and rescaling put the future null vector in the form $k^{\mu }=(1,0,0,1)$, so the profile depends on $z-t$. Transversality gives four linear relations among the ten symmetric components. The four residual gauge functions satisfying $\square {\xi }^{\mu }=0$ remove the time and longitudinal components; the remaining trace can be removed by the residual transformation indicated in the question. The resulting transverse-traceless gauge isThe two arbitrary functions are the plus and cross gravitational-wave polarizations. They are the two physical degrees of freedom left after the four gauge conditions and four residual coordinate freedoms are removed.