I can't be okay with a child going through a change of gender, and that's due to what I consider to be good sense.
The human brain takes many years to be completely formed. I can't believe that a child truly knows what they want, so allowing and encouraging them to make such a drastic change can, long term, become very dangerous.
I accept that some people feel wrong in their body. That's a fact. However, they need to reach a certain degree of maturity before they're sure that hormonal treatments and irreversible body changes are the best choice.
Changing gender comes with years if therapy, surgeries to change one's appearance which, by the way will result in infertility so they need to understand and be okay with never having children, and a very important point that in my opinion should be emphasized a lot more: the hormonal treatments are for the rest of their lives. It doesn't matter how many surgeries they have and how much they might look like the other gender, until science can change their DNA, the change of gender will continue to depend on hormones.
Now, hormones have an impact in all functions of the human body so messing up with their balance can come with very heavy consequences health wise. Gender changes are a recent thing so only time will tell what the hormonal treatments will do to people in the long run. Is it worth the risk? An adult can make that choice. It is their body after all. But a child? If they're not old enough to drive or have a job, should they be making this kind of decision? I don't think so and if it was my child I wouldn't allow it.