I've traveled IAH > NRT with ANA with an injection that needed to be kept chilled, spoke to flight attendant who kept it in a refrigerator. I kept it in a travel bag (like half the size a normal toiletry bag?) and it kept well. Though this was at least 8 years ago, I feel like they'd still be accomodating.
I suggest bringing several ice packs and several copies of the doctor's note about keeping it chilled (just in case). If you can't keep the injections in the fridge per the Airline/flight attendant (most tend to be accepting), ask them to keep the ice packs in the freezer and rotate as you see fit.
Agreed with ^ txaggiemiles said. Can you bring a small Yeti cooler or even a big vacuum thermos to keep it cold? My big hydroflask could hold my syringe and ice tube/straws. Good luck!