You have purchased a Site Owner account on Twittercounter. This is not, and does not include, the Premium Basic account. So you do not have access to the features of the Premium Basic account.
The invisible widget code should be right below the normal code for the widget when you click on "Generate Preview and Code" on this page http://twittercounter.com/pages/twitterwidget. Do you see this piece of code? And if so, does it work when you paste it into your website?