Mhh the thing is I was not able to get it working by the paper diagramms and node assignement. What I did was change the z88 file and imported only one tet into FreeCAD, moved and turned it in a way it is in space like an own in FreeCAD created tet10. Node1 FreeCAD was Node1 Z88. Than I changed the node order during import to the same as FreeCAD. Same I did for export. It seams to work. See attached the official z88 tet10 examples and your small file exported from FreeCAD.
i had a closer look at the z88structure-File and it is absolutely correct. Also your mesh-pictures look correct, so your import/export-functions seem to work!