Historically I have taken an airline from PER to a connecting point in SEAsia, then ticketed with AA to the states (now more likely to be CX).
There are a number of points that would be less expensive than SIN. KUL/SGN/TPE come to mind. CMB if you don't mind traveling a bit out of the way for very nice Biz fares.
In any case if I were going to fly 3K PER-SIN, I'd almost certainly book it separately to ensure I got a ~ and a few points and SCs with QF. UL code shares with MH are now an interesting alternative to get to CMB.