Context. SDSS J2222+2745, at z = 0.489, is one of the few currently known lens clusters with multiple images (six) of a background (z = 2.801) quasar with measured time delays between two image pairs (with a sub-percent relative error for the longer time delay). Systems of this kind can be exploited as alternative cosmological probes through high-precision and accurate strong lensing models.Aims. We present recent observations from the Multi Unit Spectroscopic Explorer (MUSE) on the Very Large Telescope (VLT) and new total mass models of the core of the galaxy cluster SDSS J2222+2745.Methods. We combine archival multi-band, high-resolution imaging from the Hubble Space Telescope (HST) with our VLT/MUSE spectroscopic data to securely identify 34 cluster members and 12 multiple images from 3 background sources. We also measure the stellar velocity dispersions of 13 cluster galaxies, down to HST F160W = 21 mag, enabling an independent estimate of the contribution of the sub-halo mass component to the lens total mass. By leveraging the new spectroscopic dataset, we build improved strong lensing models.Results. The projected total mass distribution of the lens cluster is best modelled with a single large-scale mass component, a galaxy scale component, anchored by the VLT/MUSE kinematic information, and an external shear component. The best-fit strong lensing model yields a root mean square separation between the model-predicted and observed positions of the multiple images of 0'29. When analysing the impact of systematic uncertainties, stemming from modelling assumptions and used observables, we find that the resulting projected total mass profile, the relative weight of the sub-halo mass component, and the critical lines are consistent, within the statistical uncertainties. The predicted magnification and time-delay values are, instead, more sensitive to the local details of the lens total mass distribution, and vary significantly among lens models that are similarly good at reproducing the observed multiple image positions. In particular, the model-predicted time delays can differ by a factor of up to similar to 1.5.Conclusions. SDSS J2222+2745 is a promising lens cluster for cosmological applications. However, due to its complex morphology, the relatively low number of secure "point-like' multiple images, and current model degeneracies, it becomes clear that additional information (from the observed surface brightness distribution of lensed sources and the measured time delays) needs to be included in the modelling for accurate and precise cosmological measurements. The full VLT/MUSE secure spectroscopic catalogue presented in this work is made publicly available.