We prove well-posedness results for time-inhomogeneous stable-driven McKean-Vlasov stochastic differential equations with a convolution drift where the interaction kernel belongs to some Lebesgue-Besov space. The novelty of this work is that we manage to go below -1 in space regularity for such a kernel. This is achieved under additional smoothness conditions on the initial data and divergence conditions on the kernel. The proof heavily relies on a suitable product rule in Besov space. We prove smoothing properties of the law, which allow us to have the drift in a Lebesgue-Besov space of non-positive regularity.