Hey everyone!
Update: The Brat is alive! So after doing all of the electrical tests and finding that everything checked out fine, myself and a friend who knows a little more about carburetors than I do decided to take the carburetor cover off of the carburetor body itself and found that the seat for the needle (I believe that is the correct terminology) was backed all of the way out, preventing the proper fuel flow. Now with it properly adjusted, the Brat is running great. Thank you all so much for your help.