Format the panel and admin date and clock in the stored application timezone

The panel snapshot now carries the admin-saved `timezone` ("" when unset);
the coordinator threads it onto the household due-today sensor, and the
Party Board and Quest Log cards format their kicker, header and dock
date and clock in it, falling back to Home Assistant's zone only when
none is stored. The admin app loads the stored zone once
(AppTimezoneProvider) and uses it for completion clocks, History day
grouping, clocks and lateness, and "today" in Schedule, History and new
task defaults; Settings publishes a saved zone app-wide at once.

Resolves review finding TZ-001 (task 7e173881).

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

frontend/tests/render-party-board.mjs

@@ -3,13 +3,15 @@
  *
  * Usage: node render-party-board.mjs < spec.json
  *
- * The spec JSON carries {bundle, tag, config, hass, url?, click_plate?};
+ * The spec JSON carries {bundle, tag, config, hass, url?, click_plate?, now?};
  * stdout receives one JSON object {plates, tappable, location, html} where
  * plates is the ordered plate list the shadow DOM renders (name +
  * present/away/unknown kind), tappable the plate names rendered as tappable
  * <button> plates, and location the window pathname after the optional plate
- * tap. The Python panel tests drive this harness so the assertions run
- * against the exact bundle HACS ships.
+ * tap. With now: <ISO instant> the card's clock is pinned there
+ * (argument-less new Date() and Date.now() return it; the process TZ
+ * decides its local wall time). The Python panel tests drive this harness
+ * so the assertions run against the exact bundle HACS ships.
  */
 import { readFileSync } from "node:fs";
 import { pathToFileURL } from "node:url";
@@ -39,6 +41,23 @@ globalThis.window = dom.window;
 // jsdom Event must shadow it for the dispatched event to be delivered.
 globalThis.Event = dom.window.Event;
 
+// Pin the clock: argument-less construction and Date.now() read `pinned`,
+// every other Date use is the real class.
+if (spec.now !== undefined) {
+  const pinned = new Date(spec.now).getTime();
+  const RealDate = Date;
+  class PinnedDate extends RealDate {
+    constructor(...args) {
+      super(...(args.length === 0 ? [pinned] : args));
+    }
+    static now() {
+      return pinned;
+    }
+  }
+  globalThis.Date = PinnedDate;
+  dom.window.Date = PinnedDate;
+}
+
 await import(pathToFileURL(spec.bundle).href);
 
 const element = document.createElement(spec.tag);