I can’t seem to follow your links to see what your simplex density looks like so I’m not sure if there’s a direct way to code the RNG.
But, as was recently pointed out to me, you can always use rejection sampling to turn a given rng into one for your desired distribution. How efficient that will be depends on how expensive the densities (known rng and your distribution) are to compute and how tightly you can surround your density with a multiple of the density for which you know the rng.