I find the OP's post confusing, but the salient points are:

1. for infants in their own seats, AA may offer a discounted infant rate available on some routes (very unusual) but generally, either there is no infant fare or it'll turn out that the infant fare only functions as a discount relative to full coach, so AA.com just quotes the lowest fare available even if it's not specifically for an infant. That makes sense right? If full coach is 2K, the infant in seat fare is 1600, but you can buy an adult ticket for 1200, that's what AA.com will quote you even if it's an infant who's going to be in that seat.

2. lap child fee is 10 percent of the full-fare of the cabin. This cannot be quoted by AA.com. Since you're talking about 1200 bucks or so R/T, that's pretty clearly Y, so there's no way that the lap child fee could credibly be the same amount (unless that full Y ticket was valued at 12K, which is impossible unless you're going to the moon or something.) Call AA and ask specifically for the lapchild fee.

3. Many here would be concerned about your child's safety. Rather than rehashing that debate, at the very least if you choose to lapchild your infant, you should look into several products that are out there designed to keep the baby as safe as possible (essentially attached to you at all times). Many would say even those products are not safe enough.

I hope this helps.