Follow-through on FN-5048: the four residual fixed 100ms sleeps in the
websocket badge integration tests (3 subscription-establishment waits + 1
cross-instance pub/sub propagation wait) are replaced with deterministic
awaits on the server-side WebSocketManager subscription:changed event and the
shared pub/sub message event. Removes ~400ms of unconditional real-time waiting
and closes the ordering races the sleeps papered over.
Fusion-Task-Id: FN-5048
60bfb66c8c