How to do boolean operations on meshes for procedural cutting of geometry! (see comments)

Hi, you seem to know a lot about runtime mesh editing. I've got a few questions, sorry if it's all a bit complicated. tl;dr part is bolded.
You say that "The way slicing works is by trying to intersect every triangle from one mesh with the other." This means that the slicing mesh must be able to find an identical set of triangles in the slicee mesh, right? (though in your case, I think you said that you're generating a slicing mesh that matches the surface of the slicee on the fly)
In my game, I want to give the player the ability to make indentations in specific slicer shapes that don't match up with the geometry of the slicee. I assume this would mean editing the geometry of the slicee before beginning the slicing operation, which seems difficult, but doable. Do you know how I would go about doing that? (Also I'll be creating geometry for the bottom of the hole and stuff, but I think I can figure out that part on my own...). Basically, I have a whole bunch of triangles that aren't in any sort of grid (it's the surface of an icosphere. there will never be any naturally occurring possible matches for a square slicer mesh), and I want to prepare the underlying geometry so I can use an adaptation of your slicing technique to make square indentations :]

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