If it's flat-out not available due to licensing changes, then a Copy from one device to another is the only way to go.
If you go with USB, you'll probably need multiple flash drives or HDDs. A 16GB flash drive has about 14.3GB usable. A 32GB flash drive (or larger HDD) will present a full 16GB usable MU when configured. If you can get your hands on enough drives, USB is viable.
Have you checked with Game to see if you can transfer your DLC in-store when trading in the old console? If you do the exchange during off hours (like 11am on a Wednesday or something) perhaps they wouldn't mind you doing some copying. My local GameStop has a TV set up to test trade-ins. Maybe your Game does, too. Then you'd just need the transfer cable.