Floss: Fix Rust bluetooth logging on initialization

When applying Rust log level on initializing BluetoothLogging, we can't
directly call |is_debug_enabled| since |self.initialized| is not yet
set.

This refactors some of the functions to fix the issue.

Bug: 371889111
Tag: #floss
Test: mmm packages/modules/Bluetooth
Test: Verified manually
Flag: EXEMPT, Floss-only changes

Change-Id: I2fcaf73ea120d3a7e98edf7ef5bfde1b00f59c2c