= 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 $\Box\xi^\mu=0$ 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> is
$$
h_{\mu\nu}=
\begin{pmatrix}
0&0&0&0\\
0&f_+&f_\times&0\\
0&f_\times&-f_+&0\\
0&0&0&0
\end{pmatrix}(z-t).
$$
The two arbitrary functions are the plus and cross <gravitational wave polarization>[gravitational-wave polarizations]. They are the two physical degrees of freedom left after the four gauge conditions and four residual coordinate freedoms are removed.
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