Pure Chance gets you on the right track with his advice. You really need to check out the Things We Have Learned section of STTM. The information there will be right up your alley. You should also check out this section:
You probably don't need the battery of tests he listed as optional--I haven't seen a woman be able to successfully treat herself with those since they are highly variable. You're mostly working with Thyroid tests (which include ferritin/iron) and cortisol (which you do need!). Get the other stuff if you can, but if insurance and costs are a concern, then you aren't doing yourself a great disservice, especially since you have a demonstrated thyroid disorder.
You're right, your doc is wrong...he would be right if he said it is your body's way of getting rid of T4 (not T3) because it is overproducing...but according to you, he didn't. When you have Hashi's your body is switching between hyper and hypo production constantly...the T4 and TSH levels you demonstrated in this labwork were merely a snapshot at what was going on at the time...The RT3 levels, which have a longer half life, tell the longer term story in that you are not converting enough T4 to T3.
You need to talk to your doctor about switching over to a T3 only medicine. This has to be dosed more frequently (multiple times per day) but will give you direct benefits. Your goal is upper 20% of the Free T3 range (and elimination of symptoms). STTM has some good info on T3 meds and natural dessicated thyroid.
Another possible reason for RT3 pooling is poor cortisol. Since you do have a thyroid disorder, this is far less likely the cause, but should be investigated nonetheless (especially since you have trouble sleeping). 8 AM cortisol test is good, but a 4x daily saliva test would be the best (I think you can get these through Labcorp now).
You can also get an idea of your thyroid and adrenal output by using a temperature chart. Take your temp immediately upon waking (before getting out of bed, drinking water, anything) and record it. Then take it 3 more times throughout the day, spaced 3 hours apart starting 3 hours after getting up. Do this for 4 or so straight days.
If your temps are varying wildly from day to day, this indicates a possible cortisol imbalance. If temps are constant, but low then your cortisol is probably ok but thyroid is now. You want constant and higher (approaching 98.6 degrees). This is a good test when you are on thyroid and cortisone to check your response to meds as well...