if it's easier for you to keep it by you, then I would get a small cooler and an ice pack. so then after she is done, you can place in the cooler with the ice pack and it will keep it cold until she wants the rest.
Giving her whole millk, she is around a year old, and she shouldn't be needing to have anything at 1 in the morning. By now she should be able to go the entire night without eating. You might try one night, not giving it to her and see if she falls back asleep. But if she is in a growth spurt, she may need it.
I try not to leave milk out very long, just becuase i don't want it to go bad or sour tasting. I figure it's expensive as it is, and so i really watch it so it doesn't get wasted