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>

api/routes_panel.py

@@ -21,6 +21,7 @@ panel payload), and returns the documented JSON shape:
     {
       "today_iso": "2026-09-22",
       "cycle_day": 2,
+      "timezone": "Pacific/Auckland",
       "children": [
         {
           "child_id": 1,
@@ -45,7 +46,9 @@ panel payload), and returns the documented JSON shape:
 active children ``ORDER BY sort_order, id``); ``state`` is the
 documented panel spelling ``open`` | ``completed`` (§1 of
 design/ENTITIES-AND-SERVICES.md); ``cycle_day`` is today's 1-based day
-in the first scheduled child's custody cycle (0 when no child has one).
+in the first scheduled child's custody cycle (0 when no child has one);
+``timezone`` is the application timezone the admin stored (``""`` when
+unset), which the panel cards format their visible date and clock in.
 The same :class:`PanelSnapshotResponse` Pydantic model is the route's
 ``response_model``, so the OpenAPI document carries the shape too.
 """
@@ -126,12 +129,14 @@ class PanelSnapshotResponse(BaseModel):
 
     ``today_iso`` anchors the payload's day; ``cycle_day`` is today's
     1-based day in the first scheduled child's custody cycle (0 when no
-    child has a schedule); ``children`` is in the household's sort
-    order.
+    child has a schedule); ``timezone`` is the stored application
+    timezone (``""`` when unset); ``children`` is in the household's
+    sort order.
     """
 
     today_iso: str
     cycle_day: int
+    timezone: str
     children: list[PanelChildPayload]
 
 
@@ -218,6 +223,7 @@ async def panel_snapshot(request: Request) -> PanelSnapshotResponse:
     return PanelSnapshotResponse(
         today_iso=snapshot.today_iso,
         cycle_day=snapshot.cycle_day,
+        timezone=snapshot.timezone,
         children=[
             PanelChildPayload(
                 child_id=child.child_id,