That sounds interesting. ok. It looks fun. If I get the problem like what you said, I will make collision model with small upper cylinder and large holes at the bottom, that is seperated with visual model 
Or, I could think of a set of collision meshes that are activated only when the distance approaches. That can be change the shape by moving each small parts of collision.
As the distance increases, the upper pin becomes smaller and the lower hole becomes larger. As the distance gets closer, the size difference between the two decreases. To do this, you will need to use a combination of collision meshes of different shapes and sizes. For example, if you combine several boxes to surround a cylinder, you can use them as a hole that couls be changed its radius.
And, with proper trick or logic, maybe you could change the thousands things to dozens. Usually I handle problems in this way. several sphere or boxes can be used to mimic more complex shape.
You can consider to use physics functionality of engine, but even there, maybe you would need to aproach like this. But there seems that there are more various options.
Sometimes changing the approach direction of thinking could be helpful. In your imagine, how that “sensitivity setting” work? What is the logic that “sensitivity setting” use?