Converting between coordinates systems is always a nightmare.
Unfortunately we don't have any experience with AFNI and very limited knowledge of SPM, we might not be able to help you much in your project. But I can at least try to help you in evaluating if there is any issue in what Brainstorm exports.
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Volume approach: make sure that what you get out of Brainstorm makes sense:
Overlay the .nii obtained with Brainstorm with an MNI template in MRIcron (safer option to test transformation matrices and coordinate systems than Brainstorm or SPM). -
Surface version: Try to replicate what is done in the SPM12 export tutorial (using the dataset from the introduction tutorials: deviant condition, baseline=[-20,-15]ms, active=[85,95]ms)
https://neuroimage.usc.edu/brainstorm/ExportSpm12
(update SPM12 before testing)
I've just tested it and it works as expected, leading me to think there is no problem with the .gii files generated with Brainstorm:
Perhaps AFNI/SUMA does not recognize that file type, so we moved on to SPM12.
The export to a SPM12 surface creates two .gii files: the cortex surface and a texture on this surface with the source values (which is indeed NOT a surface .gii file, it does not contain vertices or faces). This is explained in the tutorial. Make sure you didn't get mixed up.