It is perfectly possible. Gel ice is an cargo industry standard way of transport temperature sensitive products around the world e.g. pharmaceuticals. Judt get guidance locally on how much to use based on others successful transportation. Customer service won't know much about gel ice hence the confusion answer. Cargo can also go in temp controlled holds in the plane but that option isn't available for checked luggage

TBH check in won't ask what's in the box and as none of the items are dangerous there is no requirement for you to declare it, they'll just want to see it is well packed and will survive the transit. You might ask for fragile tags to reduce the pounding it will receive.

I have some experience with this on the cargo side so PM if you want any more help.