Recent learning-based path planners use neural networks to process visual map representations and approximate heuristics for classical search algorithms, yielding near-optimal paths with reduced search effort. However, these methods are tied to the shortest-path objective implicit in their supervision, which limits their flexibility to accommodate alternative criteria. We introduce FlexPath, a two-stage framework that decouples feasibility from preference. In Stage 1, we use imitation learning to acquire a task-independent spatial prior over feasible paths from visual map inputs. In Stage 2, differentiable Path Shape Objectives (PSOs) adapt this prior toward task-specific criteria without relearning path structure, requiring only efficient objective-level adaptation. A single pretrained model can be adapted to multiple objectives. For shortest-path planning, FlexPath reduces search effort on TMP by 14.3% compared to the state-of-the-art TransPath, while also finding lower-cost paths on average and demonstrating strong zero-shot generalization across three unseen domains. For obstacle clearance with minimum clearance distance 2, it achieves 96.8% full obstacle avoidance while maintaining low search cost. The framework further extends to semantic-aware avoidance and waypoint guidance via objective-level adaptation, and remains compatible with classical planners at inference time. Data and code are available at https://github.com/FraunhoferIVI/FlexPath.