Running the additional Free testing could show a few issues happening with a conversion issue of the hormones with your meds.
It may also if the tests come out abnormal - clue your doctor in on maybe look at the possibility of the adrenals or other glands that could be interferring with your thyroid medication.
It may be extremely helpful to see if the free testing could be done on you along with any other testing on deficiencies whether mineral/vitamin or hormonal related.
Doubling your med dose two days in a row sounds like a bad plan anyway. I'd get away from that routine. Maybe a double dose on Sunday and a double dose on Wednesday.
Why didn't your doctor just change your scrip? Two extra doses of 88mcg is an extra 176mcg per week, right? 9x88=792mcg per week. 7x112mcg=784mcg.
Why didn't your doctor just write you a scrip for 112mcg?
Your Endo should be running a Free T4 along with the TSH.
A Free T3 test would be nice, too, but not absolutely neccesary.
You could see if there are any defeciencies you can discuss with your doctor like a low Vit B or Magnesium.
I had that same situation and found I was magnesium deficient.
The high TSH can cause some thyroid swelling - from a thyroid stanpoint there is not much more to do besides getting the TSH normalized.