Efficient algorithm for computing equally distributed points in polytope?
This is an answer just to record a solution to the problem.

Given the polytope, I triangulate it without introducing new vertices. Then I generate for each simplex in the resulting simplicial complex many points in the interior with my second approach. Every coefficient vector will give distinct points because the vertices of the simplices are affinely independent. To implement this solution one can use Sage together with TOPCOM, for example.

Depending on the chosen triangulation, the dimension and and number of vertices this solution can actually be quite inefficient.