This is one of those situations where timing just works against you. China visa rules are super strict about jurisdiction, so the LA rejection makes sense even though it’s incredibly frustrating. They don’t care where you’re physically located, only where you legally reside.
Having your sister submit in NYC and mail it back sounds risky this close to Feb 16, unless you’re 100 percent sure they can process it fast enough. At this point, an agency is probably the only realistic option. They’re expensive, but they know how to push urgent cases and sometimes have same or next day slots you can’t access yourself.
China visas are not something to gamble with at the last minute. If the trip is important, I’d bite the bullet and use an agency immediately. Otherwise, there’s a real chance this turns into a cancelled flight.