I’ve been flying Newark to various spots in Asia in the same circumstances for several years now. . First two flights I asked the FA if they could store my small med bad (about the side of the amenity kits) in the fridge. Both times they politely refused saying they are not allow to store or even come in contact with passengers medications.
I was a bit concerned, but it kept well on its own.
Now I just put the medicine vial in one of those soft insulated lunch bags that have gel inside that you freeze in advance. Then I wrap that bag in a shirt and put it in the middle of my carry-on. It keeps fine. So far longest trip was EWR-NRT-MNL-USU (Coron). Over 40 hours door to door. On arrival the insulated bag was soft, but still chilly.
just don’t keep opening the bag to check on it during the flight/layovers. An undisturbed bag is a good insulator.
And btw, I’ve never had problems with this at security checkpoints either.