refactor(web): secure Pico websocket access behind launcher auth

- stop exposing the raw Pico token to the frontend
- add /api/pico/info for non-secret Pico connection metadata
- proxy /pico/ws through the launcher with same-origin and dashboard auth checks
- inject the upstream Pico websocket protocol server-side
- update frontend chat connection flow and Vite websocket proxy path
- refresh related docs and tests

web/backend/middleware/launcher_dashboard_auth.go

@@ -218,6 +218,10 @@ func validLauncherDashboardAuth(r *http.Request, cfg LauncherDashboardAuthConfig
 }
 
 func rejectLauncherDashboardAuth(w http.ResponseWriter, r *http.Request, canonicalPath string) {
+	if canonicalPath == "/pico/ws" {
+		http.Error(w, "unauthorized", http.StatusUnauthorized)
+		return
+	}
 	if strings.HasPrefix(canonicalPath, "/api/") {
 		w.Header().Set("Content-Type", "application/json")
 		w.WriteHeader(http.StatusUnauthorized)

web/backend/middleware/launcher_dashboard_auth_test.go

@@ -40,6 +40,7 @@ func TestLauncherDashboardAuth_AllowsPublicPaths(t *testing.T) {
 		{http.MethodPost, "/api/auth/logout", http.StatusTeapot},
 		{http.MethodGet, "/api/auth/logout", http.StatusUnauthorized},
 		{http.MethodGet, "/api/config", http.StatusUnauthorized},
+		{http.MethodGet, "/pico/ws", http.StatusUnauthorized},
 	} {
 		rec := httptest.NewRecorder()
 		req := httptest.NewRequest(tc.method, tc.path, nil)
@@ -160,3 +161,22 @@ func TestLauncherDashboardAuth_CookieAndBearer(t *testing.T) {
 		t.Fatalf("bearer auth: status = %d", rec2.Code)
 	}
 }
+
+func TestLauncherDashboardAuth_WebSocketUnauthorizedDoesNotRedirect(t *testing.T) {
+	cfg := LauncherDashboardAuthConfig{ExpectedCookie: "deadbeef", Token: "x"}
+	next := http.HandlerFunc(func(_ http.ResponseWriter, _ *http.Request) {
+		t.Fatal("next handler should not run without auth")
+	})
+	h := LauncherDashboardAuth(cfg, next)
+
+	rec := httptest.NewRecorder()
+	req := httptest.NewRequest(http.MethodGet, "/pico/ws", nil)
+	h.ServeHTTP(rec, req)
+
+	if rec.Code != http.StatusUnauthorized {
+		t.Fatalf("status = %d, want %d", rec.Code, http.StatusUnauthorized)
+	}
+	if got := rec.Header().Get("Location"); got != "" {
+		t.Fatalf("Location = %q, want empty", got)
+	}
+}