The possible issue with VLC for some people does not impact the function of the thyroid. It can impact thyroid hormone conversion. Thyroidmanager.org* suggests that people need a minimum of 50 grams/day of glucose for good T4( storage hormone)to T3( active hormone) conversion. With this in mind, ingesting less than 50 grams of carbs/day long term MAY negatively impact conversion.
I recall reading somewhere (can't remember where) that an issue with VLC and conversion is person specific- not impacted at all, impacted quickly with VLC, impacted after long term VLC or something in between.
That being said, other things can impact conversion as well:
1. Not eating enough
2. Low selenium. 200mcg/day is needed
3. Low Vitamin D
4. Low ferritin
* Effects of the Environment, Chemicals and Drugs on Thyroid Function – Thyroid Disease Manager
scroll down to Nutrition
The Non-Thyroidal Illness Syndrome – Thyroid Disease Manager
scroll down to Low T3 states
I think you mentioned something about feeling best with a TSH of 3. It left me wondering if your medication dosage was determined by TSH alone. If you haven't had freeT4 and freeT3 tested I suggest you have those tests run.