Periodically driven quantum dots can act as counterparts of cyclic thermal machines at the nanoscale. In the slow-driving regime of geometric pumping, such machines have been shown to operate in analogy to a Carnot cycle. For larger driving frequencies, which are required to increase the cooling power, the efficiency of the operation decreases. Up to which frequency a close-to-optimal performance is still possible depends on the magnitude and sign of on-site electron–electron interaction. Extending our previous detailed study on cyclic quantum-dot refrigerators [Phys. Rev. B 106, 035405 (2022)], we here find that the optimal cooling power remains constant up to weak interaction strength compared to the cold-bath temperature. By contrast, the work cost depends on the interaction via the dot’s charge relaxation rate, as the latter sets the typical driving frequency for the onset of non-geometric pumping contributions.