Do I need triangle information in my surface shader?

Update

The main question is: How can I pass the world space vertex positions of the triangle to the surface shader in a Unity shader. As mentioned in a comment it might be possible to pass them from a geometry shader. But I read somewhere that implementing a custom geometry shader overwrites Unitys logic to calculate shadows etc. I would add the triangle information in the Input structure. But before I change my mesh generation logic for it I would like to know if this is feasible. For this solution the vertex positions of the triangle must be constant for every pixel in a triangle and not be interpolated.

This is the original question:

I am writing a surface shader for a triangle mesh. I set a custom vertex attribute with a texture id to every vertex. Now I want the surface shader to apply the texture as seen in the following image. (Note that each color is representing a texture)

In the surface shader I need the 3 vertices that define the triangle and their texture ids. Furthermore I need to position of the pixel I am drawing.

1. If all texture ids are the same I pick this texture for all pixels.
2. If one or two texture ids differ I calculate the pixels distance to the triangle vertices and pick the texture like seen in the next image:

The surface shader needs to be aware of the pixels triangle. With this logic I should get the shading I am looking for. I am creating my mesh programmatically so I can add the triangle vertices and their texture ids as vertex attributes and pass it to the surface shader.

But I am not sure if this is feasible with how surface/vertex shaders work. Is there a relationship between the vertex and the pixel to get my custom triangle information from? Is there a better way of doing this?

I am using Unitys ShaderLab for my shaders.

Ok I think I found a solution. Thank you for the comments, they where useful.

Fist I change my grid topology to not use shared vertices. With this I can use a vertex color channel to set the texture ids.

vertexColor.r = vertexTextureId0; // Texture to use for vertex 0
vertexColor.g = vertexTextureId1; // Texture to use for vertex 1
vertexColor.b = vertexTextureId2; // Texture to use for vertex 2

I do not have to worry about interpolation because all vertices of the triangle have the same color information. Now I create a texture to look up to which vertex my pixel belongs to. This texture looks similar to the images I posted in the question. I have to transpose the UV coordinates according to the TWO or THREE case. This solution gives me the freedom to easily change the edge and make it more ragged.