For the discrete version, doesn't this boil down to converting a program that semi-computes a semi-measure into a program that "semi-samples from" a semi-measure?

Which is pretty straightforward:

We interpret the remainder of our (infinite) input as a real number y in [0,1]. Every time the measure that we're semi-computing adds an amount p of probability to one of its possible values v, we increment a variable x (which was initialised to 0) by p. If and when x finally exceeds y, we return the value v whose probability was just incremented.

(The continuous version, which is the one we actually need, looks harder, but I guess the same general idea should work. Perhaps we just repeat the algorithm above once for every bit of the output?)