Can you change your flight once you are in the country?

Don't attempt to deceive Chinese immigration unless you want to be detained, fined and banned from China in the future.

I get your point. But I would have both flights on the same day, so I'm wondering if I can get to the airport, register with both, go through immigration with the "other country" one so no problem, then board the "same country" one. I know its a bit cheeky, but is it doable?

Your plan is not “a bit cheeky”. It is completely reckless. Don’t even think about it. Chinese police have boarded aircraft and hauled passengers off for less. You really do not want to risk being detained by Chinese police, it will be extremely unpleasant for you. Anyway, with the technology, cameras and facial recognition now in place you won’t get as far as the aircraft.

You do this and it will no longer be an immigration issue, but a national security issue. Go figure.

Beside what Vincent and Joe Doe said, it's not possible what you like to do.
If you check-in on both flights, they'l lsee it on the monitor at the security check (before the security check, there's another control point where they check your ticket and passport).

And before security check, they stamp your boarding card.
If you wanna board a different flight, you have no stamp on your boarding pass and they'll for sure check your records.