This issue is indeed confusing, I should say.
My 6yo daughter will turn 7 on July 5.
Her older brother or I could share a seat with her.
Oh, in that particular case, she until March 31 can take a Shinkansen train free of regular fees at her seat which cannot be reserved or, on the lap of her brother / you at his / your reserved seat.
Miyako Odori begins on April 1, so according to your itinerary planned now at present, child rate fees should apply to her Shinkansen ride back from Kyoto; but still the cost is around 7000 yen, about 50 percent of the price in Japanese yen of a child 7-day (Ordinary-Class) Japan Rail Pass.
In Japan, there are two age standards of passengers’ age brackets by which rates of rail fees are decided.
(A) Child rate fees should apply to passengers who have reached 6 but have not reached 12; adult rate fees should apply to passengers who have reached 12.
(B) Child rate fees should apply to passengers who are at the primary school ages; adult rate fees should apply to passengers who are at the junior high school ages and who are older.
(Under the Japanese school system, one who is 6 years old when March ends should be admitted to a primary school in the following April.
For example, in April 2017, kids whose legal date of birth is during the one year from April 2, 2010 to April 1, 2011 should become primary school pupils.)
The six JR regional companies and some other major rail operators are in theory adopting the standard (A), but concerning regular fees, they are practically using the standard (B).
Also, the standard (B) is clearly adopted in the rules concerning IC transportation cards of major brands such as Suica and PASMO.
On the other hand, concerning Japan Rail Passes, which are offered jointly by the six JR regional companies, the standard (A) is used.
To your daughter, coincidentally these two age standards will make a difference until March 31.
She will be under the Japanese primary school ages until March 31, so she can travel by JR rail lines free of regular fees until that day, in accordance with the standard (B).
However, when it comes to Japan Rail Passes, she will need a child pass because she has reached 6, in accordance with the standard (A).
Anyway, a Japan Rail Pass never seems to pay off for her in this journey, unless you greatly change your itinerary.