Do user database settings affect maintenance job times for system databases?
Those changes will have no affect on the amount of time system backups take to run.

As a first step I would change the job schedules so that the Integrity checks and System backups are not running at the same time.

Also make sure that there are no other jobs running at the same time as either of these.

If you don't have monitoring in place, jump on the server when the backups are running, have a look and see what load the server is under. You can also try moving the Fulls back by an hour or so and see if the time change makes a difference.