A good thyroid doctor is willing to treat a hypothyroid patient clinically by testing and adjusting Free T4 and Free T3 as necessary to relieve symptoms, without being influenced by resultant TSH levels. Symptom relief should be all important, not just test results. Your Free T4 and Free T3 levels are too low for many people. What was your NatureThroid dosage? What symptoms were you having associated with those test results you listed?
Most people seem to take their T4 med in the morning before eating. That provides best absorption. There is no ironclad rule for taking T3 med, but many people find it best to take 1/2 in the morning and the other 1/2 after lunch in order to level out the effect as best possible. Personally I don't see a need to split into 3 doses. As you say, a night dose might affect sleep.