Solutions to a Monge-Ampère equation on the simplex

Any set of values of $u$ at the vertices of $\Delta_k$ can be attained just by adding an affine function to $u$, which does not change $M[u]$. To see that the solution of your problem is not unique, consider $u(x,y)=ax^2+a^{-1}y^2+\mathrm{(affine\ terms)}$ with $a>0$. Clearly $M[u]=4$ for any $a$.

On the other hand, Theorem 1.6.2 in the book The Monge-Ampère equation by C. Gutierréz states that there is a unique convex solution of $M[u]=\mu$ (with $M[u]$ properly understood) with prescribed continuous boundary values in a strictly convex domain $\Omega$. Without strict convexity we can't allow arbitrary continuous boundary data; it must be at least consistent with some convex function in $\Omega$. (But I don't know if that's enough). The uniqueness part holds without strict convexity; see Corollary 1.4.7 in the same book.

This is not an answer, but...

I don't know of any previous work on this, but it appears to be well worth studying. Techniques inspired by convex geometry, like those used to solve the Minkowski problem, might work. The idea would be to first solve the equation for a finite discrete measure using a piecewise linear function.

You might want to ask Luis Caffarelli.

I am also very interested in how you came to be interested in this question.