does bacteriostatic water itself need refrigerating?
Two questions that get merged.
The bac water on its own: generally stored at controlled room temp. Doesn't need the fridge. The manufacturer label is the authority and it's printed on the vial.
A peptide mixed with it: that's a different product with different requirements, usually fridge.
So "does bac water need the fridge" and "does my mixed vial need the fridge" have different answers and people ask the first meaning the second.
On how long an opened bac water vial stays usable: the preservative gives it a beyond-use period after first puncture and that's a label thing, not something to derive from first principles. Read the vial.