When I was moving to the UK, my US doctor expressed concern about me being able to sort out my HRT because she was fully aware of NHS wait times. She asked how we could ensure my treatment remained uninterrupted. I suggested that she "double" my dose on paper and write me the longest prescription she could, which she thought was a brilliant idea. So she wrote a one year prescription at double my usual dose.
Now here's the part that's relevant to this thread: she told me that if I paid for it myself, the insurance wouldn't be involved. Ergo they wouldn't kick up a fuss. Additionally, I'd get the pharmacy's pricing, which is lower than what insurance would be billed. If my memory serves me correctly, I paid $154 for two years worth of medicine.
Here's what that looked like. I went through TSA with that and wasn't even asked any questions.