Is it possible to set an email address to always arrive in Messages?

When in Japan, I can associate my KDDI AU iPhone address (myaddress@ezweb.ne.jp) with Messages. That is convenient because even email sent from regular email accounts arrive in Messages. So it is useful for some things such as work alerts.

But I'm on travel in the U.S. using an AT&T SIM with AT&T services, so my ezweb.ne.jp address appears to be inactive.

I notice I can use myphonenumber@mms.att.net and that will work, but the sender gets all strange so it's hard to see who it is coming from.

Anyway, what I was wondering is if it is possible to associate my @icloud.com or @me.com address with Messages so even email sent from regular email clients to one of those addresses always arrives in Message rather than having to check with Mail. If the sender is using Messages it is no problem. But if the sender is just ordinary email it doesn't arrive in Messages. I was wondering if there was some setting somewhere to make such messages always go to Messages.

Thanks,

doug