As advised by this answer, I'm using
gksu "deja-dup --backup"
to backup my data, including files that have root-only permissions on them.
Both of the two ways lead to the same two problems as perceived in comparison to how Deja Dup regularly works.
No report appears after backup is finished. Regular Deja Dup used to bring up a notification, saying that backup is complete, but not in this case.
Deja Dup run with elevated privileges does not memorize either my Amazon S3 secret access key or the encryption password, I have to reenter them over and over again. Running
deja-dup --backupin the regular mode always used to apply the passwords from the last successful backup provided that backup destinations match (I had "Remember password" option turned on in all cases, of course). Hence, I'm not certain if automatic backup is going to work if enable it for the root mode Deja Dup and it's gonna take about 2 days to check it.