The arm length and the isolation in space enable LISA to probe for signals unattainable on ground, opening a window to the sub-Hz gravitational-wave universe. The coupling of unavoidable angular spacecraft jitter into the longitudinal displacement measurement, an effect known as tilt-to-length (TTL) coupling, is critical for realizing the required sensitivity of picometer$/\sqrt{\rm{Hz}}$. An ultra-stable interferometer testbed has been developed in order to investigate this issue and validate mitigation strategies in a setup representative of LISA, and in this paper it is operated in the long-arm interferometer configuration. The testbed is fitted with a flat-top beam generator to simulate the beam received by a LISA spacecraft. We demonstrate a reduction of TTL coupling between this flat-top beam and a Gaussian reference beam via introducing two- and four-lens imaging systems. TTL coupling factors below $\pm 25\,μ$m/rad for beam tilts within $\pm 300\,μ$rad are obtained by careful optimization of the system. Moreover we show that the additional TTL coupling due to lateral alignment errors of elements of the imaging system can be compensated by introducing lateral shifts of the detector, and vice versa. These findings help validate the suitability of this noise-reduction technique for the LISA long-arm interferometer.

13 pages, 10 figures