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.