leaudio: Fix for one CSIP member not streaming (race)

When first set member is moving to streaming state and second set member
updates it available context types, the second member will not be
connected to the stream.

This patch fixes that.

Bug: 330376119
Test: atest bluetooth_le_audio_client_test
Flag: Exempt, regression tested with unit test, new test added
Change-Id: I70a06e693c15b8d808bb2664b942ac6e5e050166