How translate custom email programmatically?

Question

I'm on a D9 Site. I've created a module, where I define a hook_mail, a hook_theme and a preprocess for swiftmailer twig:

function my_project_breakout_rooms_theme() {
  return [
    'my_project_breakout_room_email_invite' => [
      'variables' => [
        'brand_name' => NULL,
        'brand_message' => NULL,
        'date' => NULL,
        'start_date' => NULL,
        'end_date' => NULL,
        'mail_to' => NULL,
      ],
    ],
  ];
}

/**
 * Implements hook_mail().
 */
function my_project_breakout_rooms_mail($key, &$message, $params) {
  $message['headers']['Content-Type'] = 'text/html; charset=UTF-8; format=flowed; delsp=yes';
  $message['from'] = $params['from'];
  $message['subject'] = $params['subject'];
  $message['breakout_room_title'] = $params['breakout_room_title'];
  $message['breakout_room_body'] = $params['breakout_room_body'];
}

function my_project_breakout_rooms_preprocess_swiftmailer__my_project_breakout_rooms__my_project_breakout_rooms_mail(array &$variables) {
  $variables['breakout_room_body'] = $variables["message"]["breakout_room_body"];
  $variables['breakout_room_title'] = $variables["message"]["breakout_room_title"];
}

In a custom rest resource then I send the mail:

use Drupal\Core\Datetime\DrupalDateTime;
use Drupal\rest\ResourceResponse;
use Symfony\Component\HttpKernel\Exception\HttpException;

public function post($data) {
  $params['from'] = $this->configFactory->get('system.site')->get('mail');
  $params['brand_message'] = $data["brand_message"];
  $params['subject'] = $data['br_topic'] . ' - ' . t('Appointment confirmation');
  $params['breakout_room_title'] = t($data['br_topic']);
  $params['breakout_room_body'] = $this->getBreakoutRoomBody($data);

  foreach ($data["mail_to"] as $user_mail) {
    $params["breakout_room_body"]['#mail_to'] = $user_mail;
    $result = $this->mailManager->mail('my_project_breakout_rooms', 'my_project_breakout_rooms_mail', $user_mail, $this->getUserLanguage($user_mail), $params, NULL, TRUE);
    if (!$result['result']) {
      throw new HttpException(400, 'Impossibile inviare email');
    }
  }
  return new ResourceResponse(['email_sent' => TRUE]);
}

private function getUserLanguage($user_mail) {
  $user = user_load_by_mail($user_mail);
  return $user ? $user->getPreferredLangcode() : 'it';
}

private function getBreakoutRoomBody($data) {
  $body = [
    '#brand_name' => $data['brand_name'],
    '#brand_message' => $data['brand_message'],
    '#date' => $this->formateDate($data['br_date'], 'd/m/Y'),
    '#start_date' => $this->formateDate($data['br_start'], 'H:i'),
    '#end_date' => $this->formateDate($data['br_end'], 'H:i'),
  ];
  
  $body['#theme'] = 'my_project_breakout_room_email_invite';
  return $body;
}

private function formateDate($date, $format) {
  $date = new DrupalDateTime($date, 'UTC');
  $date->setTimezone(new DateTimeZone($this->timezone));
  return $date->format($format);
}

Then in the swiftmail swiftmailer--my-project-breakout-rooms--my-project-breakout-rooms-mail.html.twig twig I print the title\body variables.

{% embed "@mytheme/mail/swiftmailer.html.twig" %}
    {% block mail_body %}
        <table border="0" cellpadding="0" cellspacing="0" width="600" class="breakout-room-body">
            <tr class="left-text">
                <td>
                    {{ breakout_room_title }}
                    {{ breakout_room_body }}
                </td>
            </tr>
        </table>
    {% endblock %}
{% endembed %}

The twig of my-project-breakout-room-email-invite.html.twig is

<div class="breakout-room-email-body">
    <p>
        {{ 'Hello'|t }},
    </p>
    <p>
        {% trans %}
            You are invited to the appoinment with Brand {{ brand_name }} on {{ date }} from {{ start_date }}
            to {{ end_date }}
        {% endtrans %}
    </p>
    <p>
        {{ brand_message }}
    </p>

    <table class="outer-table-link-wrapper">
        <tr class="text left-text" height="40">
            <td colspan="6">
                <table class="button-link" cellpadding="0" cellmargin="0" border="0" height="44" width="200">
                    <tr>
                        <td bgcolor="#E30613" valign="middle" align="center" width="200">
                            <div class="div-wrapper">
                                <a href="{{ drupal_url('/breakout', {absolute: true, query: {mail: mail_to}}) }}">
                                    {{ 'Reply'|t }}
                                </a>
                            </div>
                        </td>
                    </tr>
                </table>
            </td>
        </tr>
    </table>
    <p>{{ 'If you are not yet registered, please consider signin up before responding to the Brand\'s invitation'|t }}</p>
    <table class="outer-table-link-wrapper">
        <tr class="text left-text" height="40">
            <td colspan="6">
                <table class="button-link" cellpadding="0" cellmargin="0" border="0" height="44" width="200">
                    <tr>
                        <td bgcolor="#E30613" valign="middle" align="center" width="200">
                            <div class="div-wrapper">
                                <a href="{{ url('simplesamlphp_auth.saml_login') }}">
                                    {{ 'Sign Up'|t }}
                                </a>
                            </div>
                        </td>
                    </tr>
                </table>
            </td>
        </tr>
    </table>
</div>

The email is successfully send, but I can't find a way to send the translated version of that, the email are always send in Italian - the site has Italian as default language and English as other.

I've already found some questions on the argument, but they couldn't solve my problem: trying to use the ConfigOverrideLanguage orrenderRoot or both don't change anything for me.