Hi Travis; thank you for the very detailed write-up.
Basically it would seem that, in any situation, a 403 is issued when you try to push a NuGet package?
In this case, it's likely that something is intercepting the request and issuing a 403. I think that a "nuget push" does a PUT, which is often blocked by things like WebDAV. So I would make sure it's disabled, and that the load balancer isn't disabling it.