I believe there is something called the **Metropolis Algorithm** The volume of your simplex (without the linear constraint) is $\frac{1}{n!}$ and so rejection sampling is terrible.

So instead you do a reflecting random walk in your space and the limiting distribution is uniform in the polyhedron.

If you want to know rigorously **why** the Metropolis-Hastings algorithm works, maybe you can try [What do we know about the Metropolis-Hastings algorithm?](http://www.math.cornell.edu/~lsc/Whatdwk.pdf) by Diaconis and Salff-Lacoste.

Oh!  I found it... [Gibbs/Metropolis algorithms on a convex polytope](https://arxiv.org/abs/1104.0749) ... but these very complicated.  Just remember the phrase "detailed-balance".