I would strongly suggest booking the lap infant at the same time so you are all on the same PNR. As a simple example (and it is hidden somewhere in these threads), I needed to add a lap infant part way thru a DONE, it was a nightmare (luckily only a one-off): QR point blank refused to do it (had to remove wife from the booking and buy separate tickets with miles after a denied boarding, that I knew was 90% likely to happen); The cost of adding a lap infant for some one-way BA/AA/CX across ocean flights exceeded the cost of the DONE. The intra-America flights are free though on AA.

Basically, these are complicated tickets, trying to add a lap infant is not necessarily going to happen. Rather than use the tool, and as someone who prefers self-service normally, I would recommend that the AA RTW desk is just excellent and can take care of this all. If you have your route already planned out you can make a booking via phone in less than 20 minutes by just reading out segment by segment.

Even with this, it isn't always smooth sailing: A recent AONE I had the lap infant added at birth, but for a couple of the airlines at check-in I needed to read out the ticket number (CX,JL,AA), as it didn't seem to 100% associate with the overall booking, though you could see the details on the PNR.