Split a mesh while using same UVs and texture position.

I'm working on a mesh splitter tool

What my tool currently does to "split" meshes: It creates 2 new objects each one with a new mesh. The new meshes have the same number of vertices and triangles as the original mesh, but only the selected triangles are rendered.

I want to keep the texture in the same position as the original but only show the part that is on the rendered triangles.
What would be a simple approach to apply the same texture from the original object into the divided parts?

Here's an image showing what I want to achieve: