This doesn't actually answer the question (how to tweak I/O priority for a backup operation) but may help significantly speed up backups and reduce their affect on other activity:
If you can write your backups to drives that are not holding the database's data and log files. With spinning disk based drives this improve performance considerably because the backup operation is not constantly causing the heads to flip between the area holding the live data (to read pages) to the area the backup is being written to (to store those pages).
This may mean adding an extra physical drive (or multiple drives so you can use RAID for redundancy) to the machine for the backups to go to, particularly if you only have one drive or array that holds everything. If you have your data and log files on different disks/arrays but don't have a third drive/array to write the backups to and are unable to add one, writing the backups to the drive with the logs is usually faster than writing them to the drive with the data files (as the active pages in the log files are far fewer except in unusual cases).