TLDR; Try this: https://4allfamily.uk/products/rxcoolers-mini-refrigerated-travel-case-for-insulin-and-medicine
I'm in a similar situation this summer. I'm travelling for around 36 hours (door to door) to a hot destination, including a 5 hour layover about 2/3rds of the way into my journey. I'm changing airlines at that point, so need to retrieve and re-check luggage, and go through security again.
I'm also away for three weeks, and will need to start a new pen part-way through, so can't just keep the meds at room temperature as otherwise it'll be more than 30 days by the time I finish the second pen.
I've got myself one of these:
https://4allfamily.uk/products/rxcoolers-mini-refrigerated-travel-case-for-insulin-and-medicine
and some of these:
https://www.amazon.co.uk/dp/B0FQCC8J3T
As well an old-school thermometer and some glass test tubes. I've been testing by filling the test tubes with water and putting the whole thing together with the ice-pack, to make sure that it doesn't freeze the water and stays cool long enough. I've also done a basic test on the electric cooler and it worked great, but I still need to test the longevity when running off a battery pack.
My current working plan is to use the ice-pack for the long first flight, as it'll still be frozen for going through security. Then during the layover I'll be retrieving my luggage and need to re-check it, so can stick the melted ice-pack in there so it doesn't upset security. I'll then switch to using the powered cooler off a battery pack for the last part of the journey.
I can also use the powered cooler when travelling between accommodations and when we're staying at places without fridges. It can also help out for the return journey - but I'm less worried about that one, as by then I'll have started the second pen and it won't matter if it comes to room temperature as it'll be used up in <30 days.
Sorry, that was long winded and my plan is completely over-engineered. 😂 However, I hope some of the info is helpful to you or anyone else that stumbles across this while trying to work out how to transport the meds on a long journey!