The two-dimensional non-oriented bin packing problem with due dates packs a set of rectangular items, which may be rotated by 90 degrees, into identical rectangular bins. The bins have equal processing times. An item's lateness is the difference between its due date and the completion time of its bin. The problem packs all items without overlap as to minimize maximum lateness Lmax. The paper proposes a tight lower bound that enhances an existing bound on Lmax for 24.07% of the benchmark instances and matches it in 30.87% cases. In addition, it models the problem using mixed integer programming (MIP), and solves small-sized instances exactly using CPLEX. It approximately solves larger-sized instances using a two-stage heuristic. The first stage constructs an initial solution via a first-fit heuristic that applies an iterative constraint programming (CP)-based neighborhood search. The second stage, which is iterative too, approximately solves a series of assignment low-level MIPs that are guided by feasibility constraints. It then enhances the solution via a high-level random local search. The approximate approach improves existing upper bounds by 27.45% on average, and obtains the optimum for 33.93% of the instances. Overall, the exact and approximate approaches identify the optimum for 39.07% cases. The proposed approach is applicable to complex problems. It applies CP and MIP sequentially, while exploring their advantages, and hybridizes heuristic search with MIP. It embeds a new lookahead strategy that guards against infeasible search directions and constrains the search to improving directions only; thus, differs from traditional lookahead beam searches.