feat(catalog): update the catalog live after an import, no page refresh

Organising imports, the Studio furni batch, the catalog totals and the
"import from a source" stats all used to need a full page reload, or at
best a router.refresh() that re-rendered the whole admin route, before
anything on screen reflected what the import had just written.

- live-catalog-merge.ts (new): pure tree and total arithmetic. Applies a
  delta of created pages, added offers and moved offers, recomputes depth
  for the touched subtree, bumps parent child counts and the item totals.
  Returns the input untouched when a delta is empty, so subscribers can
  bail out instead of re-rendering. Depth resolution tolerates a parent
  cycle in a dirty DB and still terminates, matching getTreeFlat.
- use-live-catalog.ts (new): one module-level store exposed through
  useSyncExternalStore, so every consumer shares a single instance without
  threading a provider through the admin layout. Deltas only apply to the
  "normal" catalog, so public and public_handlers trees stay separate.
  seedCatalogTotals() takes the first server value per mode and never
  overwrites it afterwards, so a later hard render cannot make the header
  totals jump backwards.
- actions/catalog.ts: organizeImportFurni now reports each group through
  the new OrganizedPageChange, carrying parentId, pageLayout, the icon,
  isNew and the per-source movedFrom counts, so the client can fold the
  result into the tree without reading the page back.
- organize-imports-dialog.tsx: drops useRouter and router.refresh(); the
  response is applied as a delta the moment the run finishes.
- studio-client.tsx: reads the tree from the store instead of freezing it
  with useState(initialTree), loads it on mount when empty, and refreshes
  it once a batch import settles. The batch is server-side and derives its
  import pages from furnidata, so that one path re-reads the tree via
  GET /api/admin/catalog/tree?mode=full rather than trusting the delta.
- studio/furni/page.tsx: stops calling getTreeFlat() and no longer passes
  initialTree; the store is the single source of truth for the rail.
- import-clone-client.tsx: tracks which items are already present, so
  present and clonable update per cloned row instead of only at the end.
- catalog-manager-dialog.tsx: seeds the totals once and renders the live
  values, so the header reflects an import that just ran.
- e2e/ui/fixtures/entry.tsx: drops the removed initialTree prop.

src/components/admin/catalog-manager/catalog-manager-dialog.tsx

@@ -12,6 +12,10 @@ import { TooltipProvider } from "@/components/ui/tooltip";
 import type { CatalogSelection } from "@/features/catalog/client/catalog-selection-url";
 import { useCatalogNavigation } from "@/features/catalog/client/use-catalog-navigation";
 import { useLatestRequest } from "@/features/catalog/client/use-latest-request";
+import {
+	seedCatalogTotals,
+	useLiveCatalog,
+} from "@/features/catalog/client/use-live-catalog";
 import { CatalogSearch } from "@/features/catalog/components/catalog-search";
 import type { CatalogSearchResult } from "@/features/catalog/domain/search";
 import type { RootTab } from "@/types/catalog";
@@ -164,6 +168,12 @@ function CatalogManagerInner({
 	totals?: CatalogStatsTotals | null;
 }) {
 	const { state, dispatch, catalogType, canEdit } = useCatalogManager();
+	// Server totals are the baseline; after an import the live store holds
+	// counts that are newer than both the prop and its 30s server cache.
+	const liveTotals = useLiveCatalog().totals[catalogType];
+	useEffect(() => {
+		seedCatalogTotals(catalogType, totals);
+	}, [catalogType, totals]);
 	const catQs = catalogTreeQs(catalogType);
 	const rootRequests = useLatestRequest();
 	const searchNavigation = useLatestRequest();
@@ -518,7 +528,7 @@ function CatalogManagerInner({
 				</div>
 			</div>
 
-			<StatsBar totals={totals} />
+			<StatsBar totals={liveTotals ?? totals} />
 		</TooltipProvider>
 	);
 }