Once you have pre-specified some simplices $S$ that must be included in your triangulation of the convex polytope $P$, what remains is the problem of triangulating a nonconvex region: $P \setminus S$. There are nonconvex polyhedra (in dimension 3) that cannot be triangulated. And it is an NP-complete problem to decide if a given nonconvex polyhedron can be triangulated, a 1992 result of Ruppert and Seidel.
If you want to nevertheless hope that your region can be triangulated, you might explore geometric bistellar flips to underlie an approach.