I did the T4 to T3 connection you describe on the flight connections bus a few days after Thanksgiving, coming into T3 at about 9:45 a.m. and there was literally no one in line at T3 security when we arrived. Reportedly however during the current holidays things were much worse. I am not sure that going from T4 to T3 you actually go through the FCC or the same security lines that you would go through coming in from T1 of T2 but flying Y you will have to go through the "regular" security lines if you try the landside transfer route.

Also for whatever it's worth BA can most likely give you a boarding pass for your LHR-JFK flight when you check in at CPH but it will be on BA cardstock and AA will want you to exchange it for one on AA stock at some point prior to getting on the plane. You can do this at the AA gate when you arrive but again, flying in Y you might leave youself a few extra minutes to get this done. The purpose of the BA pass seems to be limited to having something to show when goingthrough security.