Definitely HUACA. The agent can request that award space be opened in a case like this.
Yesterday I received a schedule change notification on a PNH-HKG-YYZ-ORD award I had booked in J (HKG-YYZ on CX metal). Ticketed flight on the last leg was canceled and the next one would've resulted in arrival >4 hours after originally planned which was a no-go. Couldn't move to an earlier flight because we already had a forced overnight in YYZ.
I knew before calling that there was no award J inventory on any CX flight to a NA gateway that day nor on any AA transpacific flight either, so I expected I'd have to fight to get J award space opened HKG-DFW. Instead, no fight at all - the agent simply offered it as an option, put me on hold for a few minutes to get everything solved, then came back to confirm. Really well handled, and exactly the resolution I expected. (Not that I'm happy about losing the CX flight, but there's a very good chance of switching to the HKG-ORD nonstop anyway once I get close enough to departure!)