How to make kinematic body collide using velocity

I need to create a kinematic object where other objects react to it like it has a velocity and angular velocity. A normal Kinematic object ignores its rigidbody's velocity and objects that hit it act like it has zero velocity, so you can't push against an object and have it react as you expect.

I need a way to make a rigidbody that is not be affected by collisions with static and dynamic objects, but still affect the objects it hits, essentially a one-way collision. Collision layers only allow me to prevent collisions both ways.

Currently the Super-Kinematic object is marked as a non-kinematic rigidbody, and am doing some math based on the object's unrestrained motion to set the velocity, which are then used by the physics engine in collisions.

The problem comes as soon as the kinematic object touches the static parts of the scene. The physics engine reacts to the object penetrating the collider by pushing it up with a lot of force, giving it a high velocity that affects any collision with other objects on the same frame.

Here's my current script. The angular velocity piece isn't working yet so I left it out. This works until the object touches a static piece of the scene, at which point, anything it touches on the same frame is likely to go flying away.

 public class KinematicPhysics : MonoBehaviour {
     
     Vector3 prevPosition;
     
     // Use this for initialization
     void Start () {
         prevPosition = transform.position;
     }
     
     // Update is called once per frame
     void FixedUpdate () {
         Vector3 dPosition = transform.position - prevPosition;
         rigidbody.velocity = dPosition / Time.fixedDeltaTime;
         prevPosition = transform.position;
     }
 }