I think both what your doctor said and what you said are two possible solutions and they can both work, you will not know until you try.
I would personally do what your doctor said, sounds a better way to me...T4 is long acting you do not necessarily need it every day, many people can take it every other day as suggested by your doctor (good doctor by the way).
I also agree with the 20mcg T3 your doctor has added, it is true that 'some' people find it difficult/slower to adapt to T3 so you can do as you suggest to start on 5mcg twice daily then after a couple of weeks you can try and add another 5mcg if you think all is going well and then the same later on.
This is a way to correct impaired T4 to T3 conversion, you do not necessarily need to stop all T4 (for a few people yes they have to).
Your fT4 in the blood will lower but this is as a result of the intake of T3 and it is not anything to worry about.
Your doctor is very good and sensible, trust him/her for now, also do not change more than one thing at a time as you can get confused as to what is working and what is not so if you do it your doctors way then give it time to know what it is doing before you 'jump the gun' and assume it is not working.
PS a word of advice, if you decide to start on 5mcg T3 twice a day rather than 10mcg twice a day you may hit a period where you start feeling worse/more hypo, anytime between one and eight weeks after you reduce your levo/T4, at that point you can consider upping your T3.
PS one thing I would 'politely argue' with your doctor is that your bloods may not look 'perfect' to some, there is research pointing that people with lower TSH can do better, by low they say just below the threshold of normal ie:0.04 to 0.4 or something like that. One ALWAYS has to bear in mind that the bloods are a MINISCULE part of the picture, the bigger picture is how you feel (ie symptoms and clinical signs)