We investigated a possible interaction between cold dark matter and dark energy, corresponding to a well-known interacting dark energy model discussed in the literature within the context of resolving the Hubble tension. We put constraints on it in a novel way, by creating new likelihoods with an analytical marginalization over the Hubble parameter $H_0$, the sound horizon $r_d$, and the supernova absolute magnitude $M_B$. Our aim is to investigate the impacts on the coupling parameter of the interacting model, $\xi$, and the equation of state of dark energy $w$ and the matter density parameter $\Omega_{m,0}$. The late-time cosmological probes used in our analysis include the PantheonPlus (calibrated and uncalibrated), cosmic chronometers, and baryon acoustic oscillation samples and the Pantheon for comparison. Through various combinations of these datasets, we demonstrate hints of an up to $2\sigma$ deviation from the standard $\Lambda$ cold dark matter model.