I accidentally clicked on Squallie's 2016 lab link instead of his ss, and since I was there, I took a look. I noticed that the last page shows his total T4 thyroid test as being 0.8. The next time you are doing labs, I'd suggest that you have them do further tests on that. You can get a breakdown of the T4 and total Tsh as well. The reason I'm mentioning this is twofold - one is that low thyroid is actually fairly uncommon in cats unless they've been radiated (like the SRT treatment for acro) or had their thyroid zapped because they were hyperthyroid. The thyroid is involved in pretty much every system of the body.
We had watched punkin's thyroid after the SRT and it showed as 0.8 for several blood tests. He had a lot of the symptoms of low thyroid but I didn't put it together because we'd done the tests, and while low normal, it was still in the normal range. When we did the breakout it was clear he had very low thyroid. Both low and high thyroid can make it difficult to control blood sugar. Hyperthyroid causes insulin resistance. I've not heard that low thyroid does as well, but it may be involved. I don't know. Punkin's hair also was falling out, although I just thought he was shedding a lot. I had started leaving my fleece in the car because it was black and so covered with punkin's orange hair that I couldn't wear it to work if it came into the house.
Here's punkin's ss tab for labs if you are interested in looking at what we did for the thyroid tests. It ended up being a bit of a big deal trying to figure out which specific thyroid tests to do. They come in various combinations from IDEXX and my vet wasn't sure what was needed. I ended up phoning Marje from the vet's examining room to try to figure out which tests were best for punkin at that point.
Anyway, perhaps you already have explored the thyroid issue, but in case you haven't, I wanted to mention my experience with it. That may explain why you're not having the success you're hoping for with the dose increase.