Rick and I looked at your data, and the problem with it is the NIFTI datasets are not properly formed. They are missing any kind of orientation information that would be contained in the sform or qform in the NIFTI header. This is true for both the mask and the functional datasets. The mask dataset also has the problem of not having properly matched voxel dimensions, it seems, because the mask has the "pixdim" field set to 1 mm^3 voxels instead of the 3 mm^3 voxels in your functional dataset. (The functional dataset also seems to have an incorrect TR between volumes of 1750 seconds, but that wasn't really part of your question.) Because the header is incorrect, the conversion to AFNI at that point doesn't help.

The best thing to do would be to get the package that writes the data to write it properly. You had mentioned MVPA, which I know nothing about, but I googled it and found they actually use the AFNI matlab library to write out AFNI format datasets by default, I believe. Using the AFNI format there may alleviate these headaches.

If you can't avoid the NIFTI format output of MVPA, then you can roughly work around the problem by changing the NIFTI header using a command like this to change the voxel dimensions:

nifti_tool -infiles l_FEF_anat.nii -mod_hdr -overwrite -mod_field pixdim "0.0 3.0 3.0 3.0 1 0.0 0.0 0.0"

The equivalent on the AFNI format file is this:

3drefit -xdel 3 -ydel 3 -zdel 3 -keepcen l_FEF_anat_mask+orig

To copy the geometry from a dataset to another, you can also do this:
to3d -prefix anat_masknii2.nii -geomparent sbrt15Visac_Run1.nii 3D:-1:0:64:64:32:l_FEF_anat.nii

All this is fairly dangerous though, not knowing the orientation at all, so left from right, for instance, is tricky.