Related: Finding shortest path on mesh
Once you have the loop, you will need to separate the mesh in two sets of triangles (assuming the mesh is connected, with no separate “islands”), using the loop as a border. I think I would do this by “flood-filling” the graph (actually, depth-first or breadth-frist does not matter) that you need for the path searches.
When you have two sets, separating them may have to be done by making two new geometries.