My Trek FX3 road bike won't fit in my CC unless I lay it down. If you remove the front wheel (which I don't like to do) it will probably fit standing up using a clamp for the fork, since the front fork will be angled down and lower the seat enough to clear the roof.
My bike shop recommended getting a trailer hitch and a rack to fit it. I'm not interested in adding a hitch though. I have an old Blackburn rack that straps to the back hatch door; it works okay for short trips.
Enjoy your CC!!!