I noticed today that the Platform Images for Windows Azure Virtual Machines has been updated to include a Windows Server 2008 R2 and Windows Server 2012 Image for October.
The great news is I re-tested installing BizTalk Server 2013 Beta on the new Windows Server 2012 Image and everything installed and is running without issues! This time I used a Medium Virtual Machine as compared to the Small to do the initial software installation. I noticed a huge difference in performance and installation time. Since I am not using the VM now, I have bumped it down to an extra-small that costs just over $0.01 per hour.
If you want to get BizTalk 2013 running before the official image is available, make sure you use the October Images. Best of luck.