Problem mapping a domain via a CNAME record

I am using a shared hosting server. In the user description it is mentioned that this is possible by having an add-on domain. The main domain of my multisite is pointing to the root folder, and I can have access to the clients DNS, so this should be possible ("This is a good method if you own them all though, its quick and easy!").
One problem I am having is that I do not have cPanel, so it is difficult to follow your examples.
I am now simulating a client with my own private domain which is registered
under a different user account, but with the same ISP as the WP MultiSite I want to map it to. At present I have a CNAME record for "www.<my own domain>" that is pointing to the main domain of the WP multisite.
In the WP multisite I have the Domain Mapping plugin installed (v 4.1.4) with sunrise.php moved to the wp-content folder and the "define SUNRISE ON" line included just after the "define DEBUG ON" line. The multisite is setup for subsites in subfolders.
When trying to map my own domain http://www.<my domain> to one of the subsites I get the Domain unavailable error message. So I am trying to pinpoint the cause of that error message. Can you assist there?