OK, that's it.
For others experiencing this issue, here's my way of solving this WITHOUT ANY HELP FROM AVAST allthough I'm paying for "support" and "security"
Unfortunately You'll need another Android device in my case a Samsung Tab 10.1 but I'll guess any Android sms-compatible device will do.
1. Restore your Avast Mobile Backup to that device
2. Uninstall Avast (this is crucial if You want useful backups in the future)
3. Install a free backup that is maintained and updated. For example "SMS Backup and Restore" but there are tons of KitKat compatible solutions out there.
4. Make a backup through that on Your restored device.
5. Upload the backup to Google Drive or Drop box.
7. Install the working backup app yove chosen during step 3 on Your KitKat phone
8. Download the file from step 5 to the phone and restore.
DONE!
Hope the Avast developers will get Mobile Backup working.
But I'm leaving for other more reliable solutions, Bye!