So I was able to solve this and am posting this for anyone else who has this same issue.
Surprisingly, or at least with my experience, very few people that I spoke with when trying to resolve this are even aware of the ability to merge more than 3 people into a call. The genius at the Apple Store took my word for it, but I wasn't able to actually prove to him that it worked on other phones because all the phones in the store I guess couldn't make real calls. I was however able to prove to him that it
didn't work...
Both the support lines for AT&T and Apple were also unaware until I urged them to try it and to their surprise they were successful. Finally, at the advice of one of the geniuses, I went to an AT&T retail store and asked them if it was an issue with my account. They checked and it was "provisioned" correctly.
No one at the AT&T store thought it was possible either, and while they were looking for the KB article I had linked above, one of the employees offered her phone so I could demonstrate. It worked on hers! We swapped SIM cards and with her card in my phone I was able to do it as well. So then she just gave me a new SIM card and now it works fine. If only I had known it was that easy...