The Kunming-Montreal Global Biodiversity Framework aims to protect 30% of terrestrial, inland water, marine, and coastal ecosystems worldwide, and ensuring that at least 30% of these areas are under effective restoration by 2030. Maintaining and restoring ecological connectivity between natural habitats and protected areas is a key feature of this target. Achieving it will require effective and inclusive spatial planning supported by appropriate decision-support tools. Most spatial planning models address budget as an objective and connectivity as a constraint, formulating problems with Steiner trees. In many real-world cases, such as landscape-scale restoration planning, this formulation is inappropriate when environmental managers seek to optimise connectivity under a budget constraint. This problem was previously addressed with Constraint Programming (CP) and graph variables, but the current approach is severely limited in terms of spatial resolution. In this article, we formalise this problem as the budget-constrained graph connectivity optimisation problem. Based on a real case study: the restoration of forest connectivity in New Caledonia, we illustrate why ``naive'' CP approaches are inefficient. In response, we provide a preprocessing method based on Hanan grids which preserves the existence of at least one optimal solution. Finally, we assess the efficiency of our approach in the New Caledonian case study.