You may want to check the FBX SDK Documentation again. I managed to find
GetElementBinormal(int index) and
GetElementTangent(int index), which are both part of the FBXGeometryBase class, which FBXMesh inherits from. Have you tried these? Because, if the documentation is correct (which, given past experiences, I tend to expect the worst, even with wealthier companies' documentation), those should be what you would need, if the data for the tangents and binormals are included with the mesh data.
To extract the data from the objects these methods return, you'll need to use methods found in
FbxLayerElementTemplate< Type >, namely
GetDirectArray () and
GetIndexArray (), unless you want to manipulate the actual array, which there are a few methods in that class for that, as well. You need to look up the inheritance tree with these things, since a lot of the functions will be generalized for multiple class types, and will operate in similar ways on a wide variety of starkly different data structures.