I translate “safek gadol” as a big safek (i.e. a big machlokes), not as “very doubtful”.
No. He writes that when there is supervision, there is no issue of Ma’achalim Assurim. the only remaining concern would be Chasnus, and on that he concurrs with the Tiferes l’Moshe that Chasnus does not apply to a Mechalel Shabbos.
Are you referring to the M”Y or the TT”V? (Both are machmir even when there is no chashash of maachalos assuros, read them again.)
You didn’t mention Kitzur Shulchan Aruch, Lechem Hapanim, Mahar”i Assad, Mahara”m Schick (who asks on the TL”M and remains “v’tzorich iyun”, so it’s not a clear psak, but he seems to lean towards issur), Chelkas Binyomin.
Bottom line: it’s a big machlokes acharonim (which is what I think R’ Moshe meant by “safek gadol”, esp. in light of the previous teshuva).