Your FT3 is pretty low. There are exceptions, but many people don't feel best until FT3 is in the top third of the range. Yours is in the bottom quarter. However, your FT4 looks really good. Your TSH is also close to the top of the currently accepted range of 0.3-3.0 (we, on this forum, consider any other range obsolete), indicating that you might need an increase in meds.
I would talk to your doctor about adding a tiny bit of a T3 med (e.g. Cytomel) to your current meds to try to nudge the FT3 up a bit.. I think you truly are in the fine-tuning stage, so I'd be pretty conservative about the dose, perhaps five to 10 mcg. (T3 is about four times more powerful than T4.) Most people who take T3 meds, split them into a couple of doses during the day. T3 is very fast-acting and neutralized by your body if not used promptly, so two doses keep the level more consistent. Often they take the first dose with their levo first thing in the morning, then take the second when it works best for them to get a metabolism boost...early afternoon?...late afternoon? whatever is best for you personally.
Hope this starts you off in the right direction.
FT3 is 2.56 (lab range 2.0-4.43), FT4 is 1.67 (lab range 0.71-1.85)
How I hope you have a finding on this! In the 80's I went through every test under the sun and don't want to start again.
I disagree with your doctor...he obviously doesn't understand fine-tuning! What are your actual FT3 and FT4 results and their reference ranges?