A few years ago I was stumped by a similar question about generating uniformly random numbers with less than ideal constraints. The helpful minds at MathOverflow solved it [0] but it wasn't something that you could guarantee would always work. (The question explains in more detail... Due to the miniscule probability of a never ending sequence of less than ideal random integers.)

I'm curious whether this technique could be applied to my original problem!

[0]: https://math.stackexchange.com/q/1273214/196899