Beyond-planarity concepts (e.g., k-planarity or fan-planarity) apply certain restrictions on the allowed patterns of crossings in drawings. It is natural to ask for the minimum number of crossings over all so-restricted drawings for any given beyond-planarity concept. Unfortunately, for most studied beyond-planarity concepts, already recognizing such graphs is NP-complete. We augment exact crossing number formulations, based on integer linear programs (ILPs), to be able to forbid crossing patterns, and discuss necessary algorithmic consequences. To this end, we first adapt crossing patterns to the context of abstract topological graphs, and propose further extensions that allow for provable polyhedral strengthenings over natural ad-hoc constraint classes. From the users’ perspective, they only need to provide such patterns (for known or novel beyond-planarity concepts). Our framework then automatically establishes the required model and separates the necessary constraints dynamically, without the need for additional case-specific implementations. If the computation is successful, our framework solves the membership problem, outputs a crossing-minimal feasible drawing w.r.t. the crossing pattern, and also generates a proof of membership/optimality that can be externally verified. In a short exploratory study, we see that the additional beyond-planar constraints do not drastically impair the practical performance and that our extended patterns are beneficial. Furthermore, we identify algorithmic subproblems (namely primal heuristics and preprocessing routines) that are in need of the ability to soundly tackle beyond-planar scenarios.