Does iCloud backup overwrite existing data?
Restoring an iCloud Backup will cause you to lose existing data. As part of the procedure to restore an iCloud Backup, one chooses Settings → General → Reset → Erase All Content and Settings. Therefore your data on your existing device is gone.

It may be possible to perform a delta restore of an iTunes backup. Make sure both devices are running the same version of iOS, then…


Backup your current device (and your new device in case anything goes wrong) using iTunes.
Using iBackupBot, open the backup you made of your old device and find the messages data in the messages container in the backup.
Use iBackupBot's restore functionality to restore this data on to your new device.