They sell milk in small, shelf-stable, single-serve packets, much like juice boxes (the ones you mentioned). I'd get those. You can also have your child drink a little water or apple juice, which should be available, to supplement quantities, if need be. If you don't want to worry about an ice pack, bring a few large ziploc bags and ask a food vendor for some ice after security.

Since you're coming this direction, it may not be an issue, but be prepared to possibly lose one of the boxes at security because they may make you open one for testing. I bring along coconut or almond milk for my son when we travel, and that's happened to us once. I opened the box, let the security guy wave his magic wand over it, and let me son drink it right then, since it was open.