Dark matter (DM) direct detection is now standing at an interesting juncture, where the Standard Model (SM) neutrino background and the upper bound on the DM signal cross section are starting to overlap in the DM mass region around 10 GeV. The neutrino floor, which defines the extent of the neutrino background, can be modified in different Beyond Standard Model (BSM) setups. We work in a simple BSM dark sector extension of the SM visible sector, where isospin-violating interactions occur naturally. In this model, both DM and neutrinos have, in general, isospin-violating (IV) interactions with nuclei, through a newly added local U(1) gauge boson $Z^{\prime }$. Depending on the choice of the model parameters, the coherent elastic neutrino-nucleus scattering (CE$\nu$NS) cross section can either increase or decrease, shifting up or down the neutrino floor in the parameter space. The same is true for the DM experimental upper bound, whose change is driven exclusively by the IV parameter $f_n/f_p$. Several scenarios are constructed, based on the interplay between the two regions and the allowed parameter space left between them, and discussed. The potential observation of solar neutrinos in DM direct detection experiments is also discussed in the context of our framework.