Boot up the game, quit the game, click menu over the Rock Band 4 icon, Manage Game, Scroll all the way to the right, Click Ready to Install and all available purchases in history will download.
EDIT: Easy Method 2, for those who can't get Method 1 to work go to the in-game store and browse songs A-Z, page by page, and if it says Purchased rather than Installed click on the song and install. It'll do so seamlessly, repeat page by page. This method takes about 20-30 minutes tops for lots of purchases if you got a rhythm going!
I realize a lot of songs aren't showing as purchased yet, but that's how I downloaded 200 in the mean time.