I can't address the low carb diet question (I'm a vegan, so low carb doesn't work) but I am on a fairly unusual thyroid medication regimen that you might want to explore with your endocrinologist.

First, some background. I had my thyroid removed about 10 years ago. While I was on synthroid (T4) alone, I continued to have hypothyroid symptoms even though my TSH was maintained at 1 or lower. I did some research and found an article in the New England Journal of Medicine that said that some patients do better when they get both T4 and T3 supplementation, because their bodies aren't able to convert T4 into T3 efficiently. So my endocrinologist agreed to do a trial run, and lowered my T4 while adding T3. The T3 drug is called Cytomel, and lasts about 8 hours in your body. I take 5 mcg of Cytomel in the morning along with 125 mcg of synthroid, and another 5 mcg of Cytomel about 2pm. It has changed my life. I don't get sluggish or tired any more, and have no hypothyroid symptoms.

You can find many articles on the internet that debunk the T4/T3 combination, but it has worked for me. My endocrinologist has only prescribed T3 for one other person, but if someone comes in complaining that they still feel awful even though their TSH is "normal", he'll consider it. (Two of my coworkers (different doctor) also take Cytomel, but only once a day, which doesn't make any sense to me since it's only good for 8 hours.)

Of course, I still have a hard time losing weight despite having my thyroid regulated properly and eating a vegan diet, but I think that's more genetics than anything else...