feat(backend,frontend): migrate data APIs from REST to native tRPC over WebSocket

Replace REST module routers with a single typed tRPC appRouter served over
/trpc (HTTP + WebSocket), and rewire the frontend to call it via
@trpc/client wsLink (browser) and httpLink (RSC data layer). Existing
/api/health + /api/metrics stay as plain Express for infra scraping.

Notable fixes surfaced by the live smoke test:
- Express 5 / path-to-regexp v8 rejects the /trpc/* wildcard route; use a
  prefix middleware that computes opts.path from the URL instead.
- nodeHTTPRequestHandler treats opts.path as the literal procedure path, so
  it is derived per-request from req.url.
- Two ws servers on one http.Server (the /ws voice socket + /trpc) collided
  and returned 400 on upgrade; both now use noServer + a manually routed
  server.on('upgrade') keyed by path.

Verified: BE tsc+biome+40 vitest green; FE tsc+biome green; live
HTTP and WebSocket calls returned real prod data.

Co-Authored-By: Claude Opus 4.5 (1M context) <noreply@anthropic.com>

services/backend/src/trpc/trpc.ts

@@ -0,0 +1,35 @@
+import { initTRPC } from "@trpc/server";
+import type { WebSocket } from "ws";
+import { createChildLogger } from "@/shared/logger/index";
+
+const logger = createChildLogger("trpc");
+
+/**
+ * tRPC context. The WebSocket transport enriches each request with the raw
+ * socket so procedures can, if needed, inspect connection metadata. The
+ * dashboard is public (no auth), mirroring the previous REST layer.
+ */
+export interface TRPCContext {
+  conn: WebSocket | null;
+}
+
+const t = initTRPC.context<TRPCContext>().create({
+  errorFormatter({ shape, error }) {
+    return {
+      ...shape,
+      data: {
+        ...shape.data,
+        // Surface a stable code + message for client-side handling.
+        code: error.code,
+        stack: undefined,
+      },
+    };
+  },
+});
+
+export const router = t.router;
+export const publicProcedure = t.procedure;
+
+// Re-export so routers can import z from one place if desired.
+export { z } from "zod";
+export { logger };