Made the paddle a floating world object instead of just a static pickup lying around the scene.
The main thing was creating a dedicated convex hull mesh for the paddle. Buoyancy calculations work better with a simplified stable hull than the render mesh, so I made a separate low-poly version just for that. Then wired the paddle up to the NetworkFloatingMesh script and set the relevant scene configuration.
After that it was just tuning mass, submerged drag, angular drag, until it felt like something you dropped in the water rather than something teleporting around.