For adults, we often say that FT4 ought to be about midrange. Babies, however, run higher levels of FT4 than adults. The FT4 range you provide doesn't appear to be age-adjusted (it looks like an adult range). Perhaps your pedi endo is adfvising you to increase because he knows babies should have higher levels??? Also, her TSH is quite high, and this usually indicates that a meds increase is in order.
If I were you, I'd ask both doctor's for the rationale for their recommendations.
Since a human body only ultimately uses Free T3 hormone. I would encourage you to ask and get a free T3 test done as well. This you can also use to compare adjustments in medication dosages.