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/snapshot.ts

@@ -3,6 +3,7 @@ import { deviceEvents as deviceEventsTable, snapshots, type Db } from "@parking/
 import { registry, type CameraDevice, type Snapshot } from "@parking/devices";
 import type { FastifyBaseLogger } from "fastify";
 import { devicesByDirection, type FlowDirection } from "./device-resolve.js";
+import { deviceEvents } from "./device-events.js";
 import type { VisionClient } from "./vision-client.js";
 
 // Camera snapshot capture, fired AFTER the barrier opens and never awaited on the
@@ -138,6 +139,9 @@ async function recognizePlate(
       })
       .run();
     logger.info(`anpr plate '${plate}' (${result.plate.confidence.toFixed(3)}) for ${identity}`);
+    // The session's entry/exit event already shipped without this (async) plate — tell the
+    // booth so it backfills the plate badge in place (no refresh). Advisory; ledger untouched.
+    deviceEvents.emitPlateRecognized({ identity, plate, direction });
   } catch (err) {
     logger.warn(`anpr recognize failed (${identity}): ${(err as Error).message}`);
   }