To answer the question of whether the C chord is "really" V of V, you need to remember one simple fact about music.
When you listen to music, you hear it progressing in time.
Therefore, analysing any chord in terms of "what comes after it" by looking at the score is just an intellectual exercise, if it has no relationship to what the music actually sounds like. A description like "V of V" only makes sense if the listener expects to be hearing common practice functional harmony, but that's not what the song is about.
The chorus starts with a Bb chord, followed by an Eb chord over the same Bb bass note. The cumulative effect of the Bb and Eb/Bb chords in fact destablilizes Bb as the "tonic" - we are used to hearing harmonic progressions that progress somewhere!
So when the C chord arrives, at last we have a progression! We don't know where we are progressing to yet, because we haven't heard what comes next, but at least we are going somewhere.
… except that actually we don't go anywhere, because the next chord is right back to Eb and then Bb again. But hey, that C was a nice surprise while it lasted.
You can replace the C with virtually any major chord get a similar effect. Try Db major, D major, Gb major, or G major, for example.
Bottom line: this isn't functional, common-practice harmony. Good luck trying to invent a functional-harmony name for a Db or Gb chord here, but they work as music. Elton John just happened to pick C instead. Maybe his backing band haven't learned Db or Gb chords yet … (just joking, of course).