Hi there,
The train ride to Miyazaki is completely covered by the JR Pass and you will be able to make some good savings over normal ticket with the JR Pass. Like you say the train ride does take a considerable amount of time, so that is something to take into mind. The trainride is very comfortable and the last part Hakata - Miyazaki is especially nice.
About the travel date, the 19th of December is not especially busy and I don't think that you should experience any trouble getting ticket for the train. The JR staff in Tokyo can reserve all the tickets for you, which means you are secured of seating all the way.
Personally would take the train over the air-plane anyway but I do understand that people would prefer a faster flight.
I hope this helps!
Daniel