The problem is that OP is ticketed JFK-REP. That is international and the infant ticket is 10% of that fare. $1,200 seems about right.

OP could purchase an AA cash ticket for JFK-SFO. That would run roughly $300 in Y and the infant ticket would be issued without charge. He could then purchase the award ticket on CX and purchase the infant ticket from CX at 10% of the lowest J fare available on CX for SFO-REP. But, that is not likely to be much of a savings because OP will then have also paid $300 for the AA ticket. In addition, he won't be able to check bags across the tickets and will have to pay bag fees for his domestic ticket (his profile does not show AA or OW status).

My guess is that the $1,200 is close to the best you are likely to do, give or take.