Please use phpmailer class and use mail.domainname as smtp with smtp authentication to send mails
I don’t think using an external mailer class is a viable option here, but you can certainly configure EE to use an external SMTP server with authentication, as suggested.
For smtp authentication, please use existing mail box like .(JavaScript must be enabled to view this email address) as user name and its password as password.
Can you please try to configure EE accordingly? If you have any questions about the SMTP server address, username or password your host should be able to help.