The problem you are having is simply a common misconception. Sending email and receiving email are not connected in any way, shape or form. Email clients (both desktop and web) give the impression that they are connected but they're not.
If you are dealing with sending email than you don't have to send through the "in box" provider's smtp servers, or their general services, you can send any email from any address through a properly configured smtp server. If your site is hosted then the web host more than likely has such an smtp server for you to use.
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