That should be fine - possibly even a tad too long a gap, 24 hours is enough.
I'm surprised by your high Free T3. I'm also surprised about the reference range for the Free T3. It is much wider, both top and bottom, than I've ever seen before.
I am not convinced by your doctor's argument. It boils down to... "I don't want to make you feel better now, because you might feel bad later".
I'm only speculating here, so take this with a pinch of salt.
Your TSH is surprisingly high, given that your Free T4 is well in range and your Free T3 is over the range. Do you know if you have autoimmune hypothyroidism? 90% of people with hypothyroidism do, so it wouldn't be unusual.
If you have antibodies and they were just gearing up for an onslaught on your thyroid, then your Free T3 could have started rising before the TSH had dropped - TSH is slower to respond to higher thyroid hormone levels in the blood than Free T3 and Free T4.
A lot of people with hypothyroidism feel better if they eat a 100% gluten-free diet. Getting tested for coeliac disease might or might not show a problem - but the test doesn't work for gluten sensitivity. You could try going g-f ruthlessly for three months. If it helps, keep it up, if it doesn't then you could go back to eating gluten again.
I'm afraid I'm struggling to come up with anything helpful, sorry.
By the way, I only noticed your reply to me by accident. If you want to reply to a particular post you have to click the Reply button on the post you are replying to. Then the person you are replying to will be notified that they have had an answer.