Add calls to Backup Manager in Preferences Files.

Backs up display order, sort order, welcome card dismissal, and
hamburger promo preferences. Other preferences are either not impacted
by user actions or store data that may not be consistent across app
versions/devices.

Bug: 32935276
Change-Id: I6e4136978640068b3a681d3507c5f9f71aedc8f2