One area that is unclear is what fare is used to calculate the 10% when the adult has bought a mileage ticket. Had the OP bought a discounted 1st class fare, then the infant fare would be 10% of THAT fare, and, hence, likely less than $1700. However, in my experience, when the adult uses mileage, the 10% is based on full fare F (or J or Y), so it can be quite the sticker shock.

What's interesting is that some airlines (like BA) have mileage awards for lap infants that are 10% of the adult award. I bet OP would happily spend 12,500 miles for that privledge. Unfortunately, AA does not do this.

Flying with an infant soon to EZE. We got him his own seat in b class. May I add that EZE is a LONG flight to have your child sit in your lap.