[RefactoredBMS] Refresh in-memory SharedPreferences instances after restore

This CL replicates ag/2284557 in RefactoredBMS (only the BMS part, the
rest is already live code).

Test: runtest -p com.android.server.backup frameworks-services
Bug: 37973765
Change-Id: I4f6cb85b879e0ee7b2b20ac998977f8c6d8fdfe0