Originally Posted by
Frayed_Yak
I would also choose the 2xJ ride.
Also a contributing factor, 2xJ also likely accumulates more qualifying/tier points if you credit to AA or BA.
Also, you get 2x lounge access, priority boarding, and extra luggage, both ways.
You'll be better taken care of in the case of IRROPs and stand a better chance for an upgrade.
If you do end up riding Y+F, do the Y leg first
First of all, welcome to FT, OP! It sounds like you've already made up your mind, but if not I'd second Frayed_Yak's (and others') good advice about doing 2xJ. In fact, one of the benefits of accruing more miles by flying 2xJ is that it could put you well on your way to a FT award F on CX down the line.
One other possible option to consider: If I understood your posts correctly, you have to keep your total cost within a certain budget. It could be that if you fly some days of the week the trip will costs less than on other days, perhaps opening up the possibility of other options (perhaps even one way in J and one way in F?)