China does allow overseas-born skaters of Chinese parentage to participate in national competitions and sometimes international ones. Of course, the Olympics are out of the question without Chinese citizenship per IOC rules, and taking on Chinese citizenship means giving up other citizenships the skater may hold (Chinese don't recognize dual citizenship). Skaters who have not reached age 18 but who were born outside China to Chinese parents may be able to get registration papers and ID's, etc. that can facilitate this process. However, at age 18, the skater becomes an adult (Chinese regulation) and has to get his own official ID as an individual. Which presumably can't be done without citizenship. I'm not sure there are any non-Chinese citizen skaters who skate for China and are at or over age 18. But I'd be interested if somebody knows of any.
Giving up one's US citizenship and taking on Chinese citizenship (if possible) has practical implications that go far beyond the limits of a competitive skating career, and I suspect it's not an avenue that Christina's parents would support. But why rock the boat? Christina currently seems to be going in the right direction, and in her senior level USA nationals debut, I expect her to place in the top 10, possibly as high as #5-6. Coupled with her JGP series results, this makes for a pretty nice season, which likely would be capped off with a Jr Worlds assignment.