Thanks for those links eamus, and thanks to the others for posting. Looks like I'll contact AA and seek, in writing, their requirements - if any such requirements exist beyond that of the phone agent I spoke to. I do think it a bit rough that the AAagent also said "your whole itinerary is likely to be cancelled without this information".

Also looks like I'll be requesting that the issuing agent (CX in this case) annotate the PNR with our full names since it seems CAPPS II can extract from the PNR pretty much anything it wants to (I am not objecting to these measures). This is consistent with the comment from newcx12345 (and I do now know what SSR is).