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/routes/ws.ts

@@ -2,7 +2,12 @@ import type { FastifyInstance } from "fastify";
 import type { Db } from "@parking/db";
 import type { LedgerEvent } from "@parking/shared";
 import { roleHasPermissions } from "../auth.js";
-import { deviceEvents, type LaneStatusEvent, type LanePresenceEvent } from "../device-events.js";
+import {
+  deviceEvents,
+  type LaneStatusEvent,
+  type LanePresenceEvent,
+  type PlateRecognizedEvent,
+} from "../device-events.js";
 import { enrichEvent } from "../event-enrich.js";
 import type { DeviceMonitor } from "../device-monitor.js";
 import type { LaneStatus } from "../lane-status.js";
@@ -65,7 +70,8 @@ type OutMsg =
   | { kind: "printer-status"; event: unknown }
   | { kind: "device-status"; event: unknown }
   | { kind: "lane-status"; lanes: LaneStatusEvent }
-  | { kind: "lane-presence"; radar: LanePresenceEvent };
+  | { kind: "lane-presence"; radar: LanePresenceEvent }
+  | { kind: "plate-recognized"; plate: PlateRecognizedEvent };
 
 export async function wsRoutes(
   app: FastifyInstance,
@@ -136,6 +142,10 @@ export async function wsRoutes(
       const offPresence = deviceEvents.onLanePresence((radar) => {
         send({ kind: "lane-presence", radar });
       });
+      // A late async plate recognition → backfill the badge on the matching feed row. Advisory.
+      const offPlate = deviceEvents.onPlateRecognized((plate) => {
+        send({ kind: "plate-recognized", plate });
+      });
 
       socket.on("close", () => {
         offLedger();
@@ -143,6 +153,7 @@ export async function wsRoutes(
         offDevice();
         offLane();
         offPresence();
+        offPlate();
       });
     },
   );