I may be wrong, but I don’t believe it’s possible to reference a completely different URL and still keep all the same content, though you can accomplish this pretty easily with URL redirects within your domain provider.
The way I’d accomplish this is by creating 2 duplicates of the site, one for mysingingbeesite.com & one for mydancingbeesite.com to act as a landing page for each, and have the site menu urls redirect there.
Hope that helps!