fix(booth): backfill the live-feed plate + make plate search work

Two booth feed fixes:

- Plate not showing until refresh. Plate recognition is async/advisory
  (snapshot.ts recognizePlate → a kind:"read" device_event keyed by the session
  identity), so it lands AFTER the entry/exit event already shipped over the WS
  without a plate; a refresh re-fetched via the bulk enrich path and showed it.
  Added a `plate-recognized` bus event (device-events.ts) emitted when the read
  is written; ws.ts forwards it; the client patchPlate(identity, plate)
  (live-store) backfills the already-rendered feed row in place and invalidates
  the Query-owned active-sessions list. No refresh.

- Plate search didn't filter. Both the live-feed (BoothScreen) and active-sessions
  (ActiveSessions) search haystacks matched the wrong field — the displayed plate
  is the ENRICHED top-level e.plate/s.plate (set by enrichEvent), not payload.plate
  (the plate is unsigned, never in the signed payload). Switched the haystacks to
  the displayed field.

Claude-Session: https://claude.ai/code/session_01Xcm6ikLgGoCxxHrxtjkk5V

apps/server/src/device-events.ts

@@ -86,6 +86,18 @@ export interface LaneStatusEvent {
   readonly exit: boolean; // true = busy (a vehicle is at the exit vicinity)
 }
 
+/** A plate was RECOGNIZED for a session AFTER its entry/exit event already shipped. Plate
+ *  recognition is async/advisory (a vision round-trip off the snapshot), so it lands a
+ *  moment after the signed event — too late for the event's own WS push to carry it. This
+ *  notifies the booth so it can fill in the plate badge on the already-rendered feed row /
+ *  active session in place, no refresh. Advisory; never touches the signed ledger. See
+ *  snapshot.ts (recognizePlate) + event-enrich.ts. */
+export interface PlateRecognizedEvent {
+  readonly identity: string; // the session identity the plate is tied to
+  readonly plate: string; // normalized plate text (trimmed, upper)
+  readonly direction: "entry" | "exit";
+}
+
 /** Per-lane RADAR presence — a vehicle-presence INPUT (loop/radar) is shorted at the
  *  entry/exit barrier, i.e. "something is in the lane vicinity" BEFORE the camera has
  *  confirmed a vehicle. Same signal that makes the physical button lamp (relay 3) blink:
@@ -168,6 +180,16 @@ class DeviceEventBus extends EventEmitter {
     this.on("lane-presence", cb);
     return () => this.off("lane-presence", cb);
   }
+
+  /** Emitted when an async plate recognition completes for a session (after its event
+   *  already shipped). Lets the booth backfill the plate badge in place. Advisory only. */
+  emitPlateRecognized(event: PlateRecognizedEvent): void {
+    this.emit("plate-recognized", event);
+  }
+  onPlateRecognized(cb: (event: PlateRecognizedEvent) => void): () => void {
+    this.on("plate-recognized", cb);
+    return () => this.off("plate-recognized", cb);
+  }
 }
 
 /** Process-wide device event bus. */