Mounjaro on Hold Luggage

Has anyone travelled with their mounjaro quikpen and put it in their hold luggage?

I know the advice is to take it as carry on but I’m travelling with my husband and children and I don’t want them to know I’m taking mounjaro so the carry on option is not going to work.

I understand the risks of my luggage getting lost and I understand that the temperature in the hold goes below freezing and can affect the medication.

My question is, has anyone done it? I’m wondering if I put it in a little travel pouch and then inside a double walled insulated cup (like a Stanley but with a sealed lid) - will this offer any form of protection at all?

I really don’t want to leave it behind because I’m away for almost a month and that’s too long to go without it 😖