We discuss an ongoing first lattice study of the doubly-charmed tetraquark $T_{cc}^+$(3875) via a three-body approach. We investigate the $DD\pi$ system in the $I=0$, $C=2$ sector, where the $T_{cc}^+$ appears as a pole in the $J^P = 1^+$ $DD\pi$ elastic scattering amplitude. The approach automatically incorporates two-body $DD^*$ and three-body $DD\pi$ effects and treats left-hand cuts due to single $\pi$ exchanges. Two CLS ensembles, X252 and X253, with pion mass $M_\pi \approx 280$ MeV, are used, and an operator set comprised of two- and three-hadron and tetraquark operators is employed to extract finite-volume energies. Additional inputs are required for the three-body finite-volume analysis, in the form of amplitudes for the $I=1$ $DD$ and $I=1/2$ $D\pi$ two-body subsystems. We present preliminary results for these subchannels and perform exploratory three-body spectra determinations for simple choices of the three-particle K-matrix $\mathcal{K}_{\rm df, 3}$, allowing a first comparison to the lattice spectrum.