If I had to do over again I would try:
1. Backup original phone to computer (if you use iCloud backup I think you have to disable find my phone to enable the manual computer backup buttons).
2. Restore new phone from computer backup you made.
3. Done (assuming the issue is only iCloud related...).
Here is how it happened to me:
1. iCloud backup of iPhone 6s
2. Full wipe of iphone 6s
3. iCloud Restore that ended in a white screen with an apple in the middle. Then a power reset to find some the call history and messages missing. Not too sure what else but I think some application data was also not restored....
4. Oh no I'm stuck!
NOTE: I tried this 2-3 times with the same outcome each time.... I also tried it on the original phone and new phone. The original iphone 6s was replaced because it had issues with locking up and camera would not focus and was swapped by apple with the same model phone.
I'm trying this right now....and have no idea what will happen.
1. iCloud backup from iPhone 6s that won't properly restore on an iPhone 6s.
2. Wipe my iPhone 5s (technically I did an iCloud backup of my 5s prior and will restore from it after I'm done since the 5s is my phone and the 6s is my wifes phone)
3. Restore iPhone 6s icloud backup backup to iPhone 5s and cross my fingers it works.
4. Backup iPhone 5s to computer
5. Wipe iPhone 6s
6. Restore iPhone 6s from computer backup that was made from the iPhone 5s.
7. Done.