This isn't an issue about acceptance at all though imo. If they were adults not being recognised by society as male or female after they'd gone through surgery it would be, but they're not and it's not; they're currently young children, not even teenagers, let alone fully grown adults, and they're about to make irrevocable life decisions (the first kid is going to be put on hormone blockers, to prevent her/him going through puberty) and are on the path towards gender surgery. Both of them.
I am not on the kids are naive bandwagon, and it is importent to get the transition stared before puberty kicks in. The 4 year old doesn't need to do anything, but dress and act as they want to. The 9 year old is getting to the point were deciding to go on blockers is important.
If they are good parents, they will let the kids be themselves. There is nothing wrong with how a kid dresses once they reach the age that they are capable, or want to dress themselves. They may want to present themselves in many ways, or they may present themselves in one specific way. It is their time to learn.
Once they reach the age of puberty, the parents need to be open and honest with there kids, regaurdless of how they present themselves. If the child has made the decision to present themselves in a way which may not fit their physical attributes, then the parents need to have a serious discussion regaudring the situation. If needed they may get in contact with a professional, as this may be something that niether parent has expereince in.
As they reach their later years of development, then genital re-assignment may be a topic. Not all transgener persons make changes here, and it is not a requirement for them to do so. If they are content with their genitals, then the the hormone blockers are all they need. If they are considering, or fully about getting gential surgery, then more discussions need to happen.
Some transgender people know who they are from the very beggining. There are many transgender people I have talked to, and many that are open in the public, that will tell you they knew from their very earliest memories that they knew something was off with their body. Just like anything, that will not be the case with everyone. Some may be uniformed, some may genuinely unsure, some may discover it later in life.
No one should be told that they don't know who they are though. Children, should not be forced to live in sadness or depression because, people tell them they don't know who they are or what they want. This is not the days of trying to push subjects like this aside, and say its a phase. We don't look at things we don't like or uncomfortable with and say God will fix this, we take it as a serious matter, and try to help. To say, someone has to suffer for 18 years, and then fight an uphill battle to become themself and relearn the world as the person they are supposed to be is extreamly cruel and cold hearted.
I know this is a relitivly new issue for many, but real people have been living this since as far back as we know. It will take time to change the ways of society. No ill feelings towards anyone that has different views.