Most stories that say "living things can't be stored" don't get terribly technical about it. You can generally assume they just mean "no live animals" and plants and other things get a pass, unless it's a specific plot point that no cellular activity of any sort can enter the storage.
Aah. That works. I imagine the rule exists so people don't conceal a squad of commandos inside a magic bag, and have a bird drop it over a castle's walls. On the other hand, if the rule is interpreted as "living things forced into the bag are killed", then the magic bag becomes the Veil of Death from Harry Potter. There would be terrible things one could do with one on a battlefield.
If one gets really picky about it, which is my own tendency when I think of these things, a freshly-killed animal is dead at a physiological level, but much of it is alive at the cellular level for quite some time.