LeAudio: Fix dumpsys race condition

Dumpsys called from another thread may break
when the service is starting or shutting down.

Bug: 269702046
Test: atest --host bluetooth_le_audio_client_test bluetooth_le_audio_test bluetooth_vc_test bluetooth_csis_test bluetooth_has_test bluetooth_test_broadcaster
Change-Id: I5edb80668b8889d55e6bdaa38fe96803f86b412e