The solution is to insert a gen4 game that has the WiFi configuration (I used Pokémon Platinum) and through the game’s menu access the WFC configuration and erase it. You then set a new connection up.
The gen5 games link the internet connection with your WFC profile, so by doing this you can only access the Mistery Gift option after deleting your save data and restarting the game. After that it works just fine.
Thank you guys for the help, because even though there were a few extra steps, you pointed me the way to go.
I currently host the DLS1, GAMESTATS, and Peerchat server for Wiimmfi. if you have any issues with leaderboards (excluding MKW!), Mystery Gifts or other in-game downloadables, or Pokémon Wi-Fi Plaza, I can try to help!