Alright, I don't think it's going to work like I want it to, but let's try. :P
Go to your Nintendo Wi-Fi settings. Select the one that works and that you normally use, then scroll down until you see "Auto-Obtain DNS". Select "No". Now right below that (Primary DNS), enter "75.89.4.56". Make sure that each of those 4 numbers are in a separate box. Now, go to the GTS in-game. Let me know what happens when you connect to the GTS. Thanks a ton. :)