There is only one case here

It is the fruit of my labour.

If this is true, then there is no fairness higher than the person 's labor staying with that person.
Child B owns the flute, and it should not be taken away.

I thought this was going to be, we have this flute, and we have several people who want it.

Further, if it is really the flute in the picture, then it should be sold and a couple cheaper flutes acquired for all the kids.

Other fishy things are Child A is good at the flute... meaning she has a flute. If this is better than hers, she could offer her older flute to Child C, maybe even offering Child B two older instruments, one of which to give to Child C?

There are many win win scenarios that could be worked out.
However, by the constraints of what was said.
Child B owns the flute and it is not up to a parent to decide to give it to someone else.