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Hello
I got the same situation with customers recently. The best I've found was to tell them:
"You're definitely right we cannot loose the .eu. So we gonna redirect it to the BESTDOMAINE.com as main website. So people will still find you and you'll get more customers and will keep the .eu."
Then you can provide some technical arguments explaining that if he sticks to this position he will loose business.
In my case it worked out 
From 50° in Dubai
Previous answers are definitely the good ones. Focus on relevance more. However you can still add a small popup message (or better module position) explaining the new redirection
Matt is right on this.
Some years you may have tried to look for satellite websites or more. Now it's definitely not recommended.
In our case we reviewed all our competitors backlinks to see if there is any good website to do guest blogging with.
But the main guideline is now is: write content, make it good and useful, promote it on social media.
It is the best methods (at least the 7/30/13)