Advance mesh alteration: Whittling wood

I am looking to make a demo to prove that this concept works. The demo will have a stick and knife where you can cut and shape the stick, making it pointy/shorter/etc.

So i'm wondering how to go about this. I know that my blade would have to be a trigger collider and I have to calculate how deep it would cut into the surface of the stick. The tricky part is when I want to segment the mesh of the stick. I could add in some mesh Boolean operators. so once the knife enters the stick it subtracts the knifes mesh from the stick, I don't like this idea because it is expensive and the stick losses mass because it is subtracted other than cut.

Another Idea would be to give the stick some code that will try and patch its surface. so I would add a plane to the knife with some attributes so the patch code sees it as a blade, and splits the surface. This would be a pain because I have to write an algorithm that can find the triangles and detect cuts. So this way it creates a new surface instead of stealing the knifes surface and inverting the triangles of it to add to the stick.

These are expensive operation either way, I will have to push as much code into a Compute Shader to help with speed.

If you guys have any idea for the best way to do this or other ways that I have not thought of please let me know. I also don't want to have to use any asset store plugins.

Thanks in advance!