I have the reason why this isn't going to work, thanks to Karen through Live Chat.

 

All my files are on a network.  It is a limitation of Visual Studio that CuteEditor needs to be developed in a local environment.

 

Therefore, I have to copy the site from our server to my local machine, and then publish it back to the server when I've finished.

 

Having done this, the editor works fine, with intellisense and code support.

 

I guess now all I need is some sort of source control which will plug into Visual Studio, Dreamweaver and whatever editing tools the rest of my company is using so that I don't overwrite their work when copying back to the server.