Observations of colliding galaxy clusters with high relative velocity probe the tail of the halo pairwise velocity distribution with the potential of providing a powerful test of cosmology. As an example it has been argued that the discovery of the Bullet Cluster challenges standard cold dark matter (CDM) model predictions. Halo catalogues from N-body simulations have been used to estimate the probability of Bullet-like clusters. However, due to simulation volume effects previous studies had to rely on a Gaussian extrapolation of the pairwise velocity distribution to high velocities. Here, we perform a detail analysis using the halo catalogues from the Dark Energy Universe Simulation Full Universe Runs (DEUS-FUR), which enables us to resolve the high-velocity tail of the distribution and study its dependence on the halo mass definition, redshift and cosmology. Building upon these results, we estimate the probability of Bullet-like systems in the framework of Extreme Value Statistics. We show that the tail of extreme pairwise velocities significantly deviates from that of a Gaussian, moreover it carries an imprint of the underlying cosmology. We find the Bullet Cluster probability to be two orders of magnitude larger than previous estimates, thus easing the tension with the CDM model. Finally, the comparison of the inferred probabilities for the different DEUS-FUR cosmologies suggests that observations of extreme interacting clusters can provide constraints on dark energy models complementary to standard cosmological tests.