So I found a solution to this issue. I know I came across this before and I'm sure I tried it, but maybe I didn't save something, or more likely, I didn't wait long enough for the change to actually have an impact on SCCM. It wouldn't be the first time my impatients has tripped me up concerning SCCM. On the General tab for the properties of the distribution point role on the server in question there is a check box, "Allow clients to connect anonymously". Checking this box appears to be all that was needed to fix this.
Is it possible to add a SCCM 2012 R2 distribution point to a server that’s already hosting a website using IIS? If so are there any special configuration issues to consider? I’ve added the DP role to the server in question and from a SCCM standpoint everything appears fine, I can distribute software to it and it otherwise seems functional. However when I try to install something from the Software Center the installation never gets past the point where the status message says, “downloading”. Eventually it will time out and say it failed. I’m somewhat new to SCCM so this may be an obvious issue, but any help would be appreciated.