It appears you have it right to me.
I don't know why the first inspector is quoting you Article 334 (NM) if you are use SE Cable (Article 338.)
His supervisor who said that Table 310.15(B)(6) only applies to single phase feeders on a 208/120V, 3ph service has it backwards. T310.15(B)(6) applies ONLY to 120/240V services, NOT to single phase feeders from a 208/120V, 3ph service.
In addition, SE is one of the conductor types specifically listed in Table 310.15(B)(16).
The inspector should have been more descriptive in his 'Article 334' requirement.........this has nothing to do with ampacity, thermal insulation, etc.
338.10(B)(4) - says SE installed indoor must comply with Art. 334 [excluding 334.80 - ampacity].
Now look at the code section the inspector mentioned: 334.12 - wrong code section.....prob. meant 334.10 - SE [like NM] is limited to Types III, IV, & V construction...the OP mentioned 5 story construction....Types III, IV & V are only allowed to go 3 stories....4 stories if fully sprinklers....with the 5th story it means the first floor [and underground parking] are Type I or II construction........NM [and thus SE] is not permitted in Type I construction.
The OP does not indicate where he is....but with this structure already to the 5th floor, chances are he is on the 2008 NEC [the code applied for inspection is based on when the plans were submitted].
Even IF he were on the 2011 - SE must be installed in raceways approved for Type I construction.....very unlikely this was done...i.e. they probably pulled the SE cable out of a Type I construction vault.
This is all supposition....but this is a very common screw up in 5-story condo's [vault in Type I construction].