Computability of Stack Cleanliness
I don't know the language in detail, but my guess is: no, stack cleanliness is not computable.

For every natural $n$, consider a Brain-Flak program $P_n$ which does the following:


take a natural $k$ as input;
without modifying the stack below, simulate Turing Machine number $n$ on empty input, for at most $k$ steps;
if the machine does not halt within $k$ steps, revert the stack to the original state, and return $0$;
otherwise, if it does halt, modify the stack below and return $0$.


$P_n(k)$ is guaranteed to halt for any inputs $n,k$.

$P_n$ will be stack-clean (on all inputs $k$, by definition) if and only if the TM number $n$ never terminates on empty input.

The latter is known to be undecidable (it's not even semi-decidable).

Hence stack cleanliness over input arity 1, output arity 1, and constant zero function is already undecidable.