Welcome to FT, BCOS!

I am not familiar with group booking, but will try to answer what I can.

You can't rely on the seatmap to determine how many seats are available, there are often tickets sold that don't have seat assignments, and some seats are held for assignment at the gate.

As you have found out, the airlines offer many different fare buckets on each flight, and the number allocated to the less expensive ones are limited. In this case, that fare is $540 and there are only two left. Since their booking engine probably won't handle selling different fare codes as a single transaction, if you increase the number it looks for the lowest fare bucket with that many seats available.

Most probably if you buy the tickets in chunks you won't reach a point where the price would be more than the TA quoted, although I doubt that you would find any with a fare rule that allows name changes. That is probably some kind of special group fare. You might get refundable tickets at the higher levels, but then you would have to book the replacement traveler at the prevailing rate, which could be higher closer to flight time. There are pages such as Expert Flyer and KVS availability tool which can tell you what the fare buckets are for a given flight. I think both have either a free mode or free trial available.

Have you tried calling AA yourself and talking to someone in their group travel dept directly? I found this number on their web site: 1-800-221-2255.