The v-number of a graded ideal is an invariant recently introduced in the context of coding theory, particularly in the study of Reed--Muller-type codes. In this work, we study the localized v-numbers of a binomial edge ideal $J_G$ associated to a finite simple graph $G$. We introduce a new approach to compute these invariants, based on the analysis of transversals in families of subsets arising from dependencies in certain rank-two matroids. This reduces the computation of localized v-numbers to the determination of the radical of an explicit ideal and provides upper bounds for these invariants. Using this method, we explicitly compute the localized v-numbers of $J_G$ at the associated minimal primes corresponding to minimal cuts of $G$. Additionally, we determine the v-number of binomial edge ideals for cycle graphs and give an almost complete answer to a recent conjecture, showing that the v-number of a cycle graph $C_n$ is either $\textstyle \left\lceil \frac{2n}{3} \right\rceil$ or $\textstyle \left\lceil \frac{2n}{3} \right\rceil - 1$.