You do not transfer the domain name to BC. Your domain host keeps the domain name. You either choose to direct all traffic to your Business Catalyst site and use BC emails or like in your case all you want to do is point the web traffic to your Business Catalyst site.
You do not need to do anything crazy like masking, or strange DNS stuff to hide page titles (which I do not get anyway).
All you need to do is change the A record to point to the BC details and then in the BC domain manager say your using external.
Here is the going live guide: