Does a reduction with $\infty$ work?
In most cases in which such a reduction is presented, the reduction also works with $\infty$ replaced by a large enough number $M$ which is at most exponential in some power of $n$. If you replace $\infty$ by $M$ then you get a bona fide instance of $\Pi$, and this instance has polynomial size (and so the reduction runs in polynomial time) since $|M| = O(n^C)$ for some $C$, where $|M|$ is the length of the binary encoding of $M$.

You are right that using $\infty$ instead of $M$ is a bit sloppy, but it often results in a cleaner description, it being understood that the "real" reduction uses some large enough finite $M$.