On Verizon (and possibly others), the ability to edit MMS APN settings is hidden. Thanks to the help of @RustyGrom and @AlvinPhilemon I was able to leverage the HOWTO: Install Windows 10 for Phones on Non-supported devices (and other hacks) method and a modified WPFlights.xml to unhide the settings. This allows Verizon Windows Phones to run fully and properly on other GSM carriers networks (AT&T in my case)
I was not able to get this to work on Windows 10 TP, as there seems to be a few bugs in the way the APN's are functioning at the moment. Use this method if you want to enable MMS APN access in Windows Phone 8.1 (and most likely 8 as well).
1. Follow instructions from the thread created by @RustyGrom, however replace the WPFlights.xml file with the file attached to this post
2. If you are having problems downloading the certificate, install the full version of Fiddler and make sure to enable HTTPS
3. At the very bottom of the Windows Insider app, you should now have an option to "Unhide MMS APN" instead of "Enable Tethering" like in the original method. Click on the "Unhide MMS APN" and go through the instructions. Once you set it and restart the phone, you should see MMS APN
Warning: This could brick your phone, so use it at your own risk. I make no warranties on the method and am not responsible if it goes wrong