Following up, calling
Physics2DServer.pin_joint_create let me create a new joint with the correct resting position via the anchor argument, which is exactly what I needed. If you add this as an answer, I'll mark it as accepted. Also, do you know how I can access that new joint's node via its ID? The creation function returns an RID, and I can call
get_id on that, but I can't figure out how to grab the actual node to work with from there.