I don't want to pay Google so I can't. How can I get support without that feature? You really need to make sure your users understand that they can only use their account for support if they have an email that they can use to send mails (= their address is owned by some 3rd party or they host their own SMTP server and have to handle spam)!
You need to immediately update your support system to reply if the mail address is not found in the database automatically!
You should own the domain of your own mail address and you should encourage that.