If anyone else is interested for future questions, I had pleasant conversations with British Airways and TSA today (no-really!) Here's their reply:

BA: They cannot guarantee that meds can be refrigerated on board. The final decision is a cabin crew decision, although they are often able to do so.

TSA: Needles and ice packs can be brought as carry-ons, so long as they are declared before screening, and one has labels on them, with a prescription.