Can you use a screenshot of the "export accounts" QR Code of Google Authenticator as a backup?
The QR code contains all your private keys for generating the 2fa login code for each of the accounts you chose to export, so yes you can use it to backup and transfer to a new device. I would test that the screen shot works properly, and be very careful with the screen shot backup.
You can even extract the actual private keys from the QR code, i.e. with this script, https://scito.ch/content/extract-secret-keys-google-authenticator-qr-export