The ticketed allowance for business class is 2 pieces max 32 kg
The OW allowance for Sapphire members allows for 2 pieces
Qantas goes beyond this for its services and allows 3 - this is not something that carries over to other carriers

You might get away without being charged for the extra piece, but be prepared to pay for it if AA elects to charge for it

If this is a r/t journey, then you can be certain that you will be charged for a 3rd piece when checking in with AA

I have heard of airlines charging at a connection point before

You will likely be ok as long as all 3 bags are checked through, but the agent that you spoke to was correct in that you do not have a 3PC allowance on the domestic leg.

If you are not happy with the risk of having to pay, keep to 2 bags and you can be sure of being ok