Wow, seems there are some sensitive people in the teaching sector. Or perhaps there is a lot of pressure.
I have only done a little teaching in China. I found it challenging. I'm sure that the background of my degrees (none in teaching) helped me do it better than if I hadn't done that study. However I found I was not a great teacher. Perhaps with further training and experience I would have improved. But perhaps some of us are just not suited to be teachers. No matter what our education.
So I can appreciate those who are good teachers. But as I have said, it is not a degree alone that makes one a good teacher. I agree it can help, but it is only one part of everything that makes a good teacher.

You should check the details carefully with your consulate or embassy. It might be something like requiring the child's birth to be registered with the consulate within 30 days to qualify for foreign citizenship. That may be a challenge if your wife is a traditional person who believes that mothers should not leave the house or even wash their hair within the first month! But you might both feel that it is a greater advantage for the child to have US citizenship. It might be slightly inconvenient for the first few years while you live in Kunming. But if you can get a visa for yourself, then you can get one for your child. As for schooling, your child won't qualify for free schooling without Chinese hukou, but you should look into schooling options. I believe that unless your budget is a real problem, there are other choices for schooling in Kunming, which may even be preferable. I am not personally familiar with the standard of public schooling here, so perhaps someone else can address that issue.
As tigertiger mentioned, your wife's eligibility to go to the US is not given. Perhaps to have a child with US citizenship would further strengthen her chances. However if both your wife and child are Chinese citizens, having both of them wanting to apply to go to the US may be an extra hurdle.
I hope you can find some other Americans who have been through similar processes to give you some specific advice. Or at least to find a helpful consultant at your consulate!

There are certain circumstances in which the Guangzhou consulate will accept documents sent by post. It would be worth ringing them to inquire.

I have sent my passport there and back twice through China Post registered mail with no problems.
I sympathise with the complicated process you are facing. I wonder if it is simpler to get your marriage legally registered in China first, perhaps making it easier to then be recognised for entry into Australia. Then in Oz you could still have a marriage celebration, even with Western-style vows if you want. I think a celebrant or minister would be fine to do that to confirm a marriage that is already legal from another country.
That is presuming that the requirements for certified documents to marry in China are actually less than to apply for the fiancee visa in Australia.
I have done it the other way around in Australia. To get my birth certificate certified to be accepted in China, I had to take it first to the Dept of Foreign Affairs & Trade in my home state for them to certify, and then take that to the Chinese consulate for them to certify! Perhaps Red Tape is the universal constant in East and West.

Actually I agree that being a good teacher is largely about temperament and experience. I hope I did not convey any arrogance about my wife holding an actual teaching qualification. I'm sure it was of some benefit to her in the classroom, but I think she learnt more helpful things in her first few years of experience than she did at university anyway!
If you encounter people who are arrogant because they hold a teaching qualification, then you have my commiserations. I'm sure there are people like that in any field.
I think the irony is the arrogance of those saying they are qualified to be a Foreign Teaching Expert in China merely because they have a degree which is unrelated to teaching. Many are good teachers, because of their temperament and experience. We know that practically employers look for degree holders, but I think we can all agree that a degree alone (whether in teaching or aeronautics) does not make one a good teacher.

So Ritta, from Dazzer's post we can see for Westerners that saving face by telling a lie can lead to the relationship being damaged or totally broken off.
How do Chinese respond when someone tells them a lie? Is their personal pride (showing off something that is false) more important than the guanxi with the other person?