You need to just call the airline that you are flying on and ask them. Every air line handles it differently, but they do make special accomodations and have special instructions for situations like yours. Any advice that you get here although well meaning, may end up misleading you, because as I said every airline is different, so what worked for them may not end up working for you and things would go much smoother for your travel if you just found the answer directly from the airline and you knew it was right and going to work. Good luck. I would not put it in your checked bags. I actually think that is the worst idea possible because checked bags get lost, delayed damaged and misplaced way too often. And you know how it goes, the one tiem you have something super important and expensive in your bag, you will be the one person on the flight whose bag gets lost! Don't check it.
My biggest worry for your problem is not the liquid icpacks, but the syringes themselves and the allowance of you to have the actual loaded syringes. If the airline has no problem with you bringing the actual syringes, no problem, I have the answer for you. Take your syringe things in ziploc bags, unless they come in a well sealed water tight packaging. Put them in a lunchbox cooler with ice as you are leaving the house. You should be able to get into the airport with that. When you get to security, dump out the ice, get through the first security check, proceed straight to the next security spot, then find a restaurant thing in the airport and get ice to refill the lunchbag. If you have to dump out the ice before you get on the plane you can and then once you get on ask the flight attendant for more ice. They can keep bringing you more ice as you need it every hour if necessary. This sounds like the best solution to me because even if you have to dump out the ice at one point you can easily and quickly get more ice. As I said though, you should check with your airline to be sure. IF it were me, I would find a way to not take the medicine with me and get it while I was in hawaii. I am sure there is some way that you can do that. If you figure it out ahead of time I am sure that it can be done, even if it takes cooridnating a doctors office, pharmaceutical company and what not, I bet it can be done.