Cannot restore iCloud backup. Stuck on "Time remaining: estimating"

I bought an iPhone 6s a month ago. On Saturday, the screen died - they said it was a bad logic board so they gave me a new one. At first, the iCloud restore seemed to work fine, but some of my app data didn't load up. This data is really important to me, so I thought I'd restore the device again and see if the data would transfer. Now, it's been stuck on "Time remaining: estimating..." FOREVER.

I've tried the hard reset a dozen times. I've tried restoring it back to factory settings. I've tried loading older backups. It doesn't matter how long I leave restoring - I've left it overnight - I end up with the message, "Cannot restore iCloud backup," no matter which backup I use. I talked to Apple Care. No one can figure out why it isn't working. The guy at the Genius Bar said it might have to do with the fact that my carrier settings were not switched over correctly when they gave me the new device. He updated it and told me to wait a day and try again. I've done that. It STILL gets stuck on that screen and won't budge!

What could possibly be going wrong?!?! Why did it work the first time and now this?? The only thing I can think of is that it's an iCloud problem... So frustrating! Please help!