Super short version - NO don't do it. Getting a Chinese Passport is relatively cheap and easy, then get a US visa for them (Visa Refusal Warning !?! note the visa is suitable for a child with an American parent - see below for "In my case:" similar "Right of Abode" UK issues) .
And the long version .....Oddly enough I am in a similar pickle at the moment. I have two kids on Chinese Passports, but the UK fees for visas now are becoming a nuisance, so I am looking at changing their nationality. The British Embassy in Beijing has a few words of wisdom on this subject. If you search government/publications/nationality-in-china, it should pop up the relevant document. I believe it is even relevant for you as a US Citizen.
In my case I have to be very careful: If you have a Chinese passport and request a UK visitor visa, it will be refused on the grounds you have Right of Abode to be a UK Citizen through your parents Citizenship and you therefore must stump up 5000 RMB odd for a special five year Visa.
The way I interpret it is you could have both passports, but they would both be essentially useless as far as I can tell. A long time ago I believe the ''third leg'' worked, using a third country for travel - in / out stamps : change passport - but I have a suspicion that the apply for a visa ''in country of origin'' has closed this loophole. If it did work it would add considerable travel hours, which would be unnecessarily stressful on a child, let alone a parent travelling with one!
Catch-22!?! If you have a Chinese Passport and a UK one, you will never "EVER" be able to apply for any UK visa in that passport, as you already have a UK Passport. Ah yes, I was thinking of getting a UK passport and then just leaving on that? NO CAN DO - you have no Entry/Exit stamp coming into China or the UK, which means you can't go In or Out of either of them..... Yes... read that again one more time very ... very .. very ... very ... very ... carefully.
The only option is to relinquish Chinese nationality entirely through your local PSB, give them your shiny new passport and ask for an Exit Visa. On return you apply for the Family Q Visa, which is renewable every year. Q visa's are a bit odd, as an adult you can/can't work - its a bit of a grey area. Should be good for a kid though. BIG NOTE on Q visa - before leaving China - you will need full translations and signatures of Notary Public on a number of documents, ALL Original documents too, no copies. I've done it many times, if you have everything in order its super simple. They say they give long Visas, but they have made me renew every single year, its a pain in the ass.
I've been here for donkeys, so I'm going to give up the kids citizenship and apply for 3 x Green Card this year. As the kids are born in China, you do get a "get out of jail free card" in that you get another chance to change back for unforeseen circumstances, but only one.
As for school, I know less about as I home school mine, but without a Chinese Family Book, you are going to be charged full whack fees for everything, not local prices. Where your located that could mean mega bucks.
Last thoughts: I have decided to change my kids passports, and where I am currently stuck, my kids names are stuck in Chinglish and don't even match my family name! It's proving another nightmare completely. SO you may need to consider this fact also.
You did not mention, but make sure your aware that if travelling solo with a child you will require documents to prove 1/ its actually your kid, and 2/ that the other parent agrees to a child leaving. I've read a few nightmare stories recently of people being caught out by this.
Remember - The child will always be eligible for a US passport in the future, no matter what you do. I would hazard a guess that the most useful reason to change Citizenship will probably arise when considering University, when an International vs Home fees comes in to play - a difference of nearly 40% extra!
I hope this helps......