Before beginning, this my jailbreak suggestion. I wouldn't know a stock/non-jailbroken method.
Get the [[App Manager]] tweak and some desktop iPhone manager such as iMazing or Xenon. App Manager is from the same team who did Filza.
App Manager is a tweak that you can backup and restore apps' data with.
App Manager saves the the data in its proprietary format ".adbk". This located in var/mobile/Library/ADManager.
The idea would be install this tweak on both the old and the new phone; backup the apps' data from the old phone (make sure you have data space since some app's data can be large); copy out and paste the ADManager folder into the new phone following the same directory; reinstall the same apps on the new phone; lastly use the tweak to restore the data.
I've used this procedure myself when upgrading from an older iPhone to a new.