Welcome.

30+ year Type 1 here. Don't worry about keeping the insulin cool - it will be just fine for the duration of the flight. The insulin vials, syringes, meter, and test strips will not (should not) draw any attention either. I never had trouble with mine, though I'm on a pump now.

The doctor letter MIGHT be useful in the event that a TSA half-wit decides that the insulin works is a security risk, but in all honesty the likelihood of that is most slim.