Fix Ladill Mail calendar sync mailbox resolution and updates.

Use the connected mail calendar mailbox when set, skip API calls when
MAIL_API_KEY_MEET is unset, and call syncRoomUpdated on room edits.

Co-authored-by: Cursor <cursoragent@cursor.com>

tests/Feature/MeetWebTest.php

@@ -740,6 +740,35 @@ class MeetWebTest extends TestCase
         ]);
     }
 
+    public function test_scheduled_room_uses_connected_mail_mailbox(): void
+    {
+        \Illuminate\Support\Facades\Http::fake([
+            'mail.ladill.com/*' => \Illuminate\Support\Facades\Http::response(['data' => ['id' => 42]], 201),
+        ]);
+
+        config([
+            'meet.calendar.mail.api_url' => 'https://mail.ladill.com/api/service/v1',
+            'meet.calendar.mail.api_key' => 'mail-test-key',
+        ]);
+
+        $this->actingAs($this->user)
+            ->post('/settings/calendar/mail', ['mailbox_email' => 'host@customdomain.com'])
+            ->assertRedirect(route('meet.settings.calendar'));
+
+        $this->actingAs($this->user)
+            ->post('/meetings', [
+                'title' => 'Custom mailbox sync',
+                'scheduled_at' => now()->addDay()->format('Y-m-d H:i'),
+                'duration_minutes' => 30,
+            ])
+            ->assertRedirect();
+
+        \Illuminate\Support\Facades\Http::assertSent(function ($request) {
+            return str_contains($request->url(), '/calendar/events')
+                && $request['mailbox_email'] === 'host@customdomain.com';
+        });
+    }
+
     public function test_host_can_create_conference(): void
     {
         $this->actingAs($this->user)