Have you googled 'cortisol thyroid connection'? There's loads there.
Mostly they are saying that inadequate thyroid hormone causes a rise in cortisol production. Basically it seems that the adrenals try to compensate for poor metabolism as a result of inadequate T3 by pumping out more cortisone. This raises blood glucose in order to optimise cell function in the 'fight or flight' situation. Then because your blood glucose is raised, insulin is raised which drives 'excess' glucose into the cells where it is converted to fat. Hey presto! - you have weight gain, and eventually insulin resistance which is the first stage of type 2 diabetes.
In young healthy subjects raised TSH was found to correlate with raised cortisol. So,, if there isn't enough T hormone about, the body will try alternative strategies to keep things going.
Now in reality our adrenals can't compensate for inadequate T3 indefinitely and give up the struggle eventually, which leads to adrenal fatigue and eventually potentially to adrenal failure or Addison's disease. In writing to your doc I would suggest you use the term adrenal insufficiency rather than fatigue. Docs have trouble understanding that just as the thyroid gland doesn't wake up one morning and refuse to work at all, the adrenals can underperform due to chronic overwork!
There is also an interesting article here
that I haven't fully absorbed yet!
When you have finished your letter, would you mind posting it here? I am sure lots of folk would be very interested.