I have booked a flight from the USA to China and am a dual national. How do I avoid China getting to know this?

There are at least four separate problems here. The following answer is speculative, but short of somebody working in Chinese immigration chiming in, you're unlikely to get a better answer.

Problem 1: Will the airline let you leave the US with a Chinese passport?

Likely answer: yes. In my experience, the US and its airlines understands the concept of multiple passports and are OK with it. If you show up at the US airport with a return trip to China and back and a Chinese passport, you'll be asked if you have a visa to return to the US: showing your American passport will easily solve this.

Now, it's possible they'll record your American passport details at this point and that those somehow filter down to somebody in China in a position to notice and care. However, on the occasions I've had to show multiple passports in the US, I don't think this has ever been the case. YMMV.

Problem 2: Will the airline in China let you board with a Chinese passport?

This is likely the most challenging bit. You'll need to show your green card or US passport to the airline at this point, and they will likely have to enter the details into the system so the US will let you in. Does this data also get piped to Chinese officials, and so fast that the immigration officer at the counter a few dozen meters away sees it? Seems unlikely, but China being a police state, who knows?

Problem 3: Will Chinese immigration let you through with an invalid green card?

Almost certainly yes. First, exit immigration is primarily concerned with whether you're a wanted criminal etc in China, whether you have a visa or not is a problem for the airline and not them. Even if they do check your green card, the US and China are not exactly best buddies, so I would be astonished if Chinese immigration had direct access to US immigration records.

Problem 4: Will the US let you in if your APIS data was with a Chinese passport?

This one there's no doubt: yes, they have to, you're an American citizen and have an absolute right to enter your own country.

So all in all, I'll go against prevailing opinion: I think your odds of pulling this off are fairly high. However, there is definitely a non-zero risk, and it would be safer to follow the process here to transit via a third country.