Safari stops working when attached to a corporate network
Corporate IT has no idea even though there's hundreds of MacBooks in
  here now.


I'm not sure how you came to that conclusion because setting a policy like "blocking Safari" wouldn't be something that decided on by rank-and-file employees; IT management knows.

There are security products on the market that go well beyond deep packet inspection where IT can can use analytics and heuristics to determine what software you're using and block it. Since it seems to take Safari a few moments to stop working, this sounds most plausible.


  Any advice?


Yes.  Talk to IT.  Ask them why they allow Chrome and block Safari (there may be a valid reason for it.)