Usually, we'd recommend installing Tableau Server *inside* the network and using a proxy server outsite the network to provide access to external users. Is that an option?
Without that, you'd need to extract the data on a computer that does have access to the network and then republish it to the Tableau Server computer manually - or write a script that does this?
I have suggested this to the network operations team. I have also recommended moving the viz's to Tableau Public and updating via the Bridge. It is in their hands now. Thank you for your reply.