I deleted everything, started over, and got nothing. It seems to be issues with the files on the server-side not my own. They're simply not downloading to me due to the checksum error.
Leandro: You seem to be right about that, but I can't tell why I'm getting the wrong files, whether I'm being redirected or not, I can't tell. But I solved this by running the launcher for private Beta, downloaded the files from that manifest, and copy/pasted them into the Homecoming folder. So it seemed to work fine for one manifest, and not the other. I'll see if this happens again in the next update.