I agree with the others that you should not have an issue with the ice packs as long as they are with the milk. Alternatively, if you do not need a huge amount of milk, you can usually find a store that sells small cartons of milk on the other side of security, and if there is a Starbucks, there is milk. You pay a premium, but it is pretty hassle free. As far as juice is concerned, we would just bring an empty sippy cup and ask the flight attendant if she would fill it for us on the plane. We did this long after our kids were too old for sippy cups because in such a confined space it was easy for them to knock over a plastic cup on a tray table. Good luck with your trip !
http://pintsizepilot.com