Ideally, at least for maintainability, it would be better to have a single site.
Obviously you can re-direct (or whatever) existing domain names to the required target URL.
Perhaps having a dual brand site such as
“Dr XXX – The Man Behind EyeMiami5 & GlassesBrickle”
or the like.
Now that’s a delicate thing of course…as the client might want to differentiate the two…
My first question is What does the client want?
After that you can discuss the most efficient way to go about it.