I've just ran into the same problem and that's how to solve it. I'm using VS 2010 Ultimate but I guess it's the same in VS 2013.
First, when you connect (and execute the sql) from the toolbar you actually request to do it on the 'Transact-SQL Editor' toolbar, which means that the toolbar handles the MS databases. That's why you ALWAYS get a connection dialog to MS databases.
If MySQL package is properly installed in VS (and apparently that's the case for you) you should:
- Right click any existing toolbar (or go to View->Toolbars) and select MySQL. This should add the MySql Toolbar.
- Click on the first button from the left (in 'MySQL Toolbar') to either connect to an existing Data Connection or create a new connection to your MySql DB. A MySql script tab will be added.
- Copy all the sql generated from the EDMX file to the 'MySql script tab'.
- Run the script from the 'MySql script tab'.
That should do it. I know there's still a little copy-paste involved but at least you don't have to leave VS.
Hope that helps
cheerio