We continue our study of the perfect kernel of the space of transitive actions of Baumslag-Solitar groups by investigating high transitivity. We show that actions of finite phenotype are never highly transitive, except when the phenotype is 1, in which case high transitivity is actually generic. In infinite phenotype, high transitivity is generic, except when |m|=|n| where it is empty. We also reinforce the dynamical properties of the action by conjugation on the perfect kernel that we had established in our first paper, replacing topological transitivity by high topological transitivity.