Okay,
First make sure that its just not a problem with data being turned off on your phone. Go to settings>general>cellular. Make sure data is on but LTE is off. If that doesn't work then what we have is a failure to communicate.
I can't believe that you would be the first person on base to deal with this but it's Korea so nothing surprises me anymore.
A Korean telecom has to register your phone in Korea for it to work. If you have voice then they probably have done this. There is no difference in frequency between your phone and the Korean one, so they are pouring the proverbial **** in your boots. I don't know how long you have been here but something I have observed is that Koreans give really good customer service. Friendly, quick, helpful, but if they don't know the answer to something that is outside the realm of usual they are quick to give an answer that it doesn't work or its not possible instead of let me find that out for you. This is where having a Korean speaker with you can be helpful. Now, where you are I would think they would be used to dealing with foreigners but I'm not sure. The only thing I can think of is that the phone registration is tied to your Alien registration card, and you guys don't have those. But I'm sure there is a rule for serving military. The bottom line is that you have to get them to follow the steps they would take as if they were setting up a Korean phone. This was a 5 minute process for me to do even before the iPhone was released here. I was told also at the time it wouldn't work. I, with the kind but firm help of my Korean wife, assured them it was possible and when they did it, it worked, has worked, and is still working now. Many of my coworkers have also brought phones from the states and no problems. If you can't get them to follow protocol your next best bet is to come into Seoul, go to a big service center where they often deal with foreigners and get it done. Find out what the rules are for you guys, there are a bunch of places around yongsan that do it. Lemme know how else I can help and thank you for your service!