Ok, that honestly sounds like a flaw in the general assembly of a docker app. What's the point in allowing users to specify ports in host mode if the joke is on the user?
Bridging I don't think is ideal as it seems to route straight to my gateway (wISP IP), as opposed to dead ending at my firewall. Why? Maybe I'm looking at this the wrong way.
That said, I changed it to bridge mode, set the port in the both the basic and advanced template locations, and now I'm greeted with a simple 404. There is no place to change it in the app, Code Server, from what I can tell so far.
EDIT: Uninstalled, reinstalled fresh. Set to bridge mode, specified port. It stuck. *facepalm* I still think we should be able to change the default port in host mode though. *shrugs*
Case closed?