SQ has a bid for upgrade system but bargains for longhaul PE=>J are scarce.
mySQupgrade: Post your bidding experiences here!
I think you're more likely to get a sub $2k upgrade offer on UA. The mileage earnings will also be considerably higher on that UA ticket. The downside is the food will be bad, service inconsistent, and the FAs won't keep the lavs clean on that 16 hour flight.