If I purchase a game on another account, do I need to redownload it again?
If you select download it should only download the license.
Like games that come with pre-order bonus, if you download them, ever noticed they are only a few KB downloads? This is because the items are already in the game, the download is just the license/activation.
So if you select to the-download it should only take two minutes at most to go threw your download then it'll finish.