If I understand you correctly, to summarise your installation -
You have connected your new server to a network, lets call it Company network A, in a corp environment.
Most of the rest of this corp are connected to a network, Company network B, and between the two is a proxy server, deciding what may or may not be visible on Company network B from the sensistive stuff on Company Network A.
Is that correct?
If so, then the proxy server needs to be configured to make the new sites visible to the greater multitude, if you like to 'publish' the websites.
From a Putty access point of view, unless there is some sort of tunneling available through the proxy, not going to happen.