Use mClientAttribution.uid directly in BasicClient constructor.

If getClientUid is overridden, the child class's implementation is
not invoked in the constructor. While not overridden anywhere currently,
this could potentially cause confusion in the future.

Bug: 369841571
Test: CameraPermissionTest
Flag: EXEMPT minor followup to mechanical refactor
Change-Id: Ieee8784e0c16c571e3dd8149c1d9ae5bd50c087f