I did some more research, and it seems like no coolers, even the fanciest ones, will hold ice longer than 5-6 days.
With that in mind, you'd have to find a way to cool the medicine box at least once every 1-2 days. Maybe replace the ice or something. I think it shouldn't be too hard, as most gas stations will sell bagged ice.
The video is somewhat crappy, but it does provide some info:
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On the other hand, If I were you, I'd plan a two weeks vacation; I would make sure to time it right between the need to inject the meds.