Integrate LiveKit SFU and harden booking workflow

components/booking/MonthCalendar.tsx

@@ -1,6 +1,7 @@
 "use client";
 
 import { useMemo, useState } from "react";
+import { berlinDateKey, formatBerlinDate } from "@/lib/calendar/timezone";
 
 export type CalendarMarker = {
   date: Date;
@@ -23,7 +24,7 @@ function startOfMonth(d: Date) {
   return new Date(d.getFullYear(), d.getMonth(), 1);
 }
 function dateKey(d: Date) {
-  return d.toISOString().slice(0, 10);
+  return berlinDateKey(d);
 }
 
 export function MonthCalendar({
@@ -62,7 +63,7 @@ export function MonthCalendar({
     return result;
   }, [month]);
 
-  const monthLabel = month.toLocaleDateString("de-DE", { month: "long", year: "numeric" });
+  const monthLabel = formatBerlinDate(month, { month: "long", year: "numeric" });
   const today = dateKey(new Date());
 
   return (