Do I understand correctly that you believe z = f(SomeRandomStuff) + no_error
You will never get good sampling but another way to put this is
z ~ Dirac(f(SomeRandomStuff))
If this is the case, it’s similar to ABC methods where you have a simulation model and are trying to match data, but you can’t match it exactly so you have to tolerate some error. I’d suggest
z ~ Normal(f(SomeRandomStuff), epsilon)
where epsilon is small, now the size of epsilon is something you can tune to compromise between the Dirac type situation (where you will never get samples at all) and the situation where epsilon is large and you’ll get samples but most of them will be too far off your target to be acceptable.