Upvote:3
According to Chinese embassy, Chinese citizens can apply for passport/travel documents at the nearest consulate, regardless of jurisdiction. (http://www.china-embassy.org/chn/lszj/zytz/t1572101.htm)
To be 100% sure, you could have your spouse contact New York Consulate or email Chinese embassy in Mandarin Chinese. They are pretty good about getting back to you.
I checked LA consulate website for travel document application requirement for your child's case. Although they also didn't mention anything about jurisdiction, proof of residency is required for the application. You will need application form, photos, child's birth certificate (original and copy), parents' passport (original and copy), parents' marriage certificate (orinial and copy), parent's proof of residency at time of child's birth (original and copy), and either you or your spouse should accompany the child to the application. (http://losangeles.china-consulate.org/chn/lszj/hzlxz/hzlxzxz/t1526294.htm)