I'd suggest running fiddler while attempting to run the script on that server, then see what exactly it's failing on so you know for sure. Download Fiddler Web Debugging Tool for Free by Telerik
I'm fairly positive your IT staff can validate specific extension types relevant to arcgis, grant your user name with special permissions, or create a new outbound rule on the firewall to achieve what you need. Is data theft a concern to your organization? If your outbound requests are over a secure connection and data theft is not a concern, then I don't think this level of security is necessary. (I know this may not be what your IT staff wants to hear)
I think this is a matter of access on the firewall that you will not be able to get around just using model builder or python. If you use fiddler and identify what https calls are failing, then you could at least report what outbound access you will need to your IT staff and see if they can grant that to you. Hope this helps.
- Steve