All the weird behavior was due to the httpRuntime between each application being different. My MVC application was set to 4.5 while my application that was setting the cookie was 4.0. Apparently there was a change in how the crypto happens behind the scenes, and therefore when the cookie came through the pipeline, it would get stripped out as ASP.NET couldn't decrypt it.
I came across this when I manually tried to decrypt the cookie by setting the name property different. That way I could access the cookie and try to dectypt, but at that point I would get an exception.
I found the following link led me in the right direction: Sharing a cookie between two websites on the same domain
By setting the compatibility mode setting on the machine key, the cookie came through just fine and could be decrypted.