I have hashi's and have been LCHF for a year. When I started LCHF I was 190 lbs and taking 150 mcg of natural desiccated thyroid (Efra's Thyroid up here in Canada). A year later I am around 156 lbs and taking the same medication. I found that the diet made no difference to my situation except when I began the diet I had an autoimmune flare-up. My guess is the diet was a shock to my body. I have a couple of other AI issues so I am not sure it was hashi's realated or not. Anyways, my flare-up included arthritis pain, fatigue, a slightly off stomach and hair loss. The usual but it was milder and of shorter duration than usual.

I haven't had my hormones tested in a year so there is a slight chance that I am leaning towards the hyper side of things what with the lower weight now, but I feel fine.

I wouldn't worry about lower insulin. Your body will still be making enough insulin to handle up to around 150g of glucose per day, even if you eat zero carb. If there really was a problem, then people who eat lower or low carb would become hypo, and that just doesn't happen. You should be fine. Just retest in a few months if you are concerned.

I can't prove it that there's a causal link, however my Hashimoto's flared up while I was on a high carb diet, and since I've been on a more moderate carb diet, my dosage of meds has never changed, in about 20 years. Nothing changed when I went low carb, at all, with my thyroid tests. Subjectively speaking, I feel more energetic on a low carb diet.