In case you only want to use apps without tracker to backup and restore your SMS, you could :
On the old phone :
- Install QKSMS from F-Droid (direct download link here),
- open it,
- set this app as the default SMS app,
- display the menu with a swipe from the left to the right,
- click on “Backup and restore”,
- click on “Backup now”,
- allow access to the internal storage if the pop up askking for this permission is displayed,
- open the “Files” app,
- save the “QKSMS” directory.
On the new /e/ phone :
- Paste this directory on the new phone, at the same place,
- open the pre-installed “Message” app,
- display the menu with a swipe from the left to the right,
- click on “Backup and restore”,
- click on “Restore”,
- click on the wanted backup,
- click on “RESTORE”.