Accepted answer
The relevant point is that sterility at the moment of reconstitution is not the same as sterility a week later after four stopper piercings.
Reconstituting with sterile water and then drawing the entire contents into separate syringes for freezing is a workaround people use; it moves the stability question rather than removing it.
More usefully, sterile water is hypotonic. Injecting a large volume of hypotonic solution causes local discomfort and haemolysis; at the 0.1 to 0.3 mL volumes typical here the effect is negligible.
The 28-day in-use period for multi-dose preservative-containing preparations is a long-standing pharmacy convention and is printed on the labelling of bacteriostatic water itself.
The caveat is that none of this makes a research-grade preparation safe for human use, which it is not, in any jurisdiction.
Refrigerate whatever you reconstitute, whichever water you used.