No , fasting would not have caused the relatively low FT4 and FT3. Weight loss and heat intolerance are generally more associated with being hyper, but they can also be related to being hypothyroid. And your lab results definitely do not indicate hyperthyroidism. I may be grasping here, but there are some additional tests I suggest for you to request from the doctor even before requesting a therapeutic trial of thyroid medication adequate to raise your FT4 and FT3 into the upper half of their ranges. Those additional tests are cortisol, Vitamin D, B12 and ferritin. The reason for the cortisol would be that if you do have a dysfunction in the hypothalamus/pituitary system, it might also affect the output of cortisol. People who are hypothyroid are often deficient in the other 3 listed, so it would be good to get those tested. D should be at least 50 ng/mL, B12 in the upper end of the range, and ferritin should be at least 100.
Your TSH level does not indicate hyperthyroidism or primary hypothyroidism. It does leave open the possibility of central hypothyroidism, which is a dysfunction in the hypothalamus/pituitary system resulting in TSH levels that do not adequately stimulate the thyroid gland to produce thyroid hormone. Your Free T4 is only at 19% of its range. Your Free T3 is only at 22% of its range. If you also had symptoms that are frequently seen with hypothyroidism those levels would usually be suspect for hypothyroidism, not hyperthyroidism. So what is it that makes you feel like you are hyper?
Just because test results are within their reference ranges does not mean they are optimal for you. The ranges are far too broad, and individuals can have different levels that work best for them. Even more important than those test results are symptoms. So please tell us about any symptoms you are having.