Reland: Enable fs-verity in background thread

The previous attempt was reverted due to CTS failure. The workaround is
implemented in the test since the API contract isn't broken.

In this reland, avoid running a custom thread by leverage the existing
background handler.

Bug: 259179666
Test: on cuttlefish, run ChecksumsTest#testInstallerSignedChecksumsUpdate
Change-Id: Ie88ae3b219e904c0cd9f85b7eb31ac21e56e36d5