I also had to change the name of the "FreeImageLibrary" string
That's how you got into this trouble. There are two DLLs involved here. FreeImage.dll is the native DLL, it contains the FreeImage_GetFileTypeU export. FreeImageNET.dll is the managed class library that contains the wrappers for the native DLL, it doesn't have that export as the exception message told you.
What you should have done was rebuild the native DLL from source so that you got the 64-bit version of FreeImage.dll. That blog post told you how to do that. And offered a download, ymmv.
It must also be copied into the build output folder so that it can be found at runtime. An easy way to do that is to add it to your project with Project + Add Existing Item and set its "Copy to Output Directory" property to "Copy if newer".