Recent point-based object completion methods have demonstrated the ability to accurately recover the missing geometry of partially observed objects. However, these approaches are not well-suited for completing objects within a scene as they do not consider known scene constraints (e.g., other observed surfaces) in their completions and further expect the partial input to be in a canonical coordinate system which does not hold for objects within scenes. While instance scene completion methods have been proposed for completing objects within a scene, they lag behind point-based object completion methods in terms of object completion quality and still do not consider known scene constraints during completion. To overcome these limitations, we propose a point cloud based instance completion model that can robustly complete objects at arbitrary scales and pose in the scene. To enable reasoning at the scene level, we introduce a sparse set of scene constraints represented as point clouds and integrate them into our completion model via a cross-attention mechanism. To evaluate the instance scene completion task on indoor scenes, we further build a new synthetic dataset called ScanWCF, which contains labeled partial scans as well as aligned ground truth scene completions that are watertight and collision free. Through several experiments, we demonstrate that our method achieves improved fidelity to partial scans, higher completion quality, and greater plausibility over existing state-of-the-art methods.