I’d agree that the song is in E Major, although the Chorus might make a bit more sense in B Mixolydian, which is a relative key of E Major (same notes, different tonal center).
The Bridge chord progression is indeed E - D - A - A - E - D - A - C, but the chords are all in root position (no inversions).
This is a fairly standard example of modal mixture, where the D and C chords are borrowed from the parallel Minor mode. You’ll find tons of these progressions on the database if you’re interested.
This is what it looks like in roman numerals:
