1TB iPhone 13 Pro Max Green Screen of Death- How to backup the data?

You don't know about it. If someone steals your iPhone you won't be notified about that either. And it doesn't matter how much you spent if you did not take advantage of the backup plans provided to you that is your issue not Apple's. You stopped backing up your data, not Apple. As I just said if you have backed up on your computer it is a trusted device. Connect your iPhone to it and you should be able to see it do a backup. Then when you get your phone repaired if the data is gone you can merely set up your iPhone from your backup.

There is no way to guarantee the data because they don't know what is actually broken. If it is not your screen but but the graphics IC on the logic board they are going to replace the device with a service replacement unit so all your data will be gone. It is actually possible that your data is already gone.

So you can try the backup to your computer or you can just take it in and take your chances. Your choice.

If your iCloud backup was full you should have either backed up to your computer and purchased more backup space, there is no telling when a device will fail, break, be stolen, etc. Apple provides the tools but the user has to take advantage of them. If you have ever backed up to your computer and trusted it as a device you can try connecting your phone to the computer and get one last backup.

If you aren't able to do this then you will have to hope that it is only a display unit problem on your iPhone and they are able to replace it without your iPhone being erased but they cannot guarantee this.