Overriding a function in a WordPress plugin

If you must use the 3rd party function – why not just reformat the returned value?

Date ($id) {
   $userActivityDate = Br::show_activity_date($id);
   date_timezone_set($userActivityDate, timezone_open('timezone You want it to be in'));
   date_format($userActivityDate, 'Y-m-d H:i:sP');
   $return = apply_filters('user_activity', $userActivityDate);
}

If the return value of the Br::show_activity_date is not directly useful in this manner, then I suggest you update your question to include what it returns, in order to provide a more complete answer.

https://www.php.net/manual/en/datetime.settimezone.php