Designing aerospace components often requires balancing conflicting objectives and constraints (aerodynamics, structural, acoustics). Traditional design processes rely on iterative trial-and-error steps until acceptable performance is achieved. While effective, this approach is resource-intensive, requires significant expertise, and often fails to explore the design space efficiently. Inverse design provides an alternative paradigm by directly generating candidate geometries that satisfy desired performance objectives, bypassing the need for iterative evaluation and modification, particularly when combined with recent advances in generative modeling. Among generative methods, diffusion models stand out for their stability and ability to generate diverse, high-quality samples. Yet their application to realistic, highly constrained optimisation remains limited due to issues such as distribution shift, constraint handling and sampling efficiency. This work addresses these challenges by proposing a score-based diffusion model complemented with tools initially developed within Cenaero's surrogate-based optimisation framework. First, constrained problems are reformulated into an aggregated single-objective formulation, used as a conditioning mechanism for the diffusion model. Second, we investigate the mitigation of distribution shift by enriching the initial database through adaptive infill sampling. The methodology is demonstrated on two representative cases. The first one concerns the aerodynamic design of the ONERA M6 transonic wing. The geometry is parameterized using the Free-Form Deformation method, and an initial database is constructed through a Design of Experiments. The diffusion model is then conditioned to generate input parameters leading to increasing lift-to-drag ratios, illustrating its capability to propose new geometries that outperform existing designs and demonstrating the effectiveness of the score-based inverse design approach. The second case uses a similar methodology to address the constrained design of high-speed propeller blades. Here, the training database for the generative model is augmented with samples obtained from a parallel surrogate-based optimisation. This demonstrates that, while diffusion models can generate promising candidates for inverse design, incorporating new samples from surrogate-based optimisation enhances the generative capacity of the diffusion model and helps mitigate issues related to distribution shift.