I just downloaded your file and tried it myself.
S3D indeed seems to have some problems with the supports showing through.
But the .stl having some issues with non manifold edges, as well.
After using the repair function and 0,2mm layer-height there remain just 4 little small points wich
in my opinion could just be ignored.
Checking on the print preview I think my Prusa I3 might print it quite well without any support.
Using the feature type preview and then carefully placing manual supports just under the overhangs
it would certainly print without any faults.
Not a real solution, I know. This really seems to be kind of a bug or at least
one of the rare designs where software or "artificial Intelligence" are at there limits.
And still, it might help if printing this one is of immediate need....