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Obviously he is thinking like a trumpet player.

Since you currently play BBb if you switched to CC you would have to learn a whole new set of fingerings (you would finger the C Major scale on the CC the same way you did the Bb salce on the BBb). Unless he wants you to switch for some other reason or your Tuba profersor wants you to switch to CC stay with BBb and just tell him you are man enough to handle the transposition. This will save you a lot of time of time debating the issue with the guy.

You really don't transpose. You just learn new fingerings for the same music. BBb, CC, Eb and F all read from the same music (in general - English Brass Band music is an entirely different beast). The horns each have different fingerings. For example: BBb player see a BBb on the sheet and plays it open (no valves); while the CC player plays that note with the first valve. Don't let ANYONE tell you that you MUST play CC in orchestra; it's just not necessary! You play on the horn with which you are most comfortable. This subject has been discussed to death and the seeming majority agree with what I have said. I DEFY any director to discern the difference in sound between two equally good horns; BBb or CC!!
I hope I have helped.

Agree totally with last post.

What type of settings are E and F tubas common in?

I hear a lot about orchestral CC players and concert band BB players, but F and E tubas seem somewhat more mysterious...