I tried something similar, many years ago, before I was diagnosed. I gained three kilos.
Sounds to me like you are Under-treated and need an increase in dose. And if that is the case, this diet is not going to help you, and might even harm you. Hypos should not go on low calorie diets because it affects their conversion, making them even more hypo.
You may also have nutritional deficiencies which will make it even more difficult for your body to use the hormone you are taking, and make you more inclined to put on weight - and a diet of this kind could accerbate the nutritional deficiencies.
You shouldn't over-do the exercise, either, because it will not make you lose weight, but will use up your T3, leaving you even more hypo. You will only be able to lose weight when your T3 level is optimal.
What would be interesting would be to see your thyroid labs - with ranges - and know how much thyroid hormone replacement you are taking, and what it is. When you are hypo, your weight-gain has little to do with what you eat, but more to do with the level of your T3.
Fibromyalgia is very often a symptom of un/Under-replaced hypo.