This might take some time to explain, but here I go :).
I created two Visual Studio 2012 projects:
So based on what I wrote in the various edits of the original post:
Now, my theory (and hope) is that this behavior is somehow specific to my development environment and won't be reproducible on the real Azure infrastructure. That said, I have come up with a workaround...
I created a simple proxy file for Firefox to redirect all traffic going to my local Azure emulator port to the IIS Express application port. This solves the problem, however I'm aware that I'm essentially bypassing the Azure emulator now, and I'm guessing I may have issues in the future because I'm doing that (e.g. if I decide to use the AppFabric cache?)!
In the mean-time I'll continue with this workaround for development and periodically deploy to my Azure staging environment for testing there.
Regardless, if anybody knows how I might solve the fundamental problem, let me know! :)
EDIT: I am no longer able to reproduce the issue using version 2.2 of the Azure SDK Tools. If you're still seeing this behaviour, try upgrading!