i would suggest a stainless steel flask aswell , put some insulation inside a pannier, an ali pannier would be better as it would reflect some heat and just apark in the shade, should be good for a day or two .
Other ideas is there an alernitive drug ? get in touch with the drug manufacturer , they may say it's ok upto a higher/cooler temp as long as it is cooled/heated before use. Try and find a good pharmacist to talk to, it's amazing some solutions they come up with.
Another spanner in your works, on the round the world trip get a note from your Dr detailing your medical problem and what drugs have to be taken and make sure that these are readily available where ever you aim to go .
If you take a load of prescribed drugs over a border you best make sure you have all bases covered , otherwise you may have free accomodation in a secure compound !!!!
There is always a solution enjoy your trip

As mentioned above we have been travailing with a 12 volt cooler that seems to do the trick of keeping wife's insulin cool. The one we took to Australia was 10x8 and tall enough for liter bottle. If we camped more than a day where we could not plug it in at night we just bought a bag of ice. Was insulated well and ice would last two days. Check out travel blog as I think there is a few pictures that will show it on her bike, if you follow it back to our South America trip you will see smaller one we used there. Where there is a will there is a way. "horizonsunlimited.com/tstories/thode"

Road Hog:scooter:

Frig
 

i have modified an al box around an engel frig 22 lbs can handle up to 30 degree angle only draws 2.5 amp max whn running .0.01 when idle put it on reat carrier looks like a bus from behind but hope it works 12 volts of course