I've thought about this a lot, driving to races and traveling and whatnot.
First choice: put the bike inside the car. Is there a reason you aren't predicting having enough space inside to do that?
Second choice: A framework or suspended net to carry the bike at the ceiling, inside the car. Keeps the floor space free, slide the bike in from the back and go.
Neither of those will impact the aerodynamic profile of the car. If you need to carry the bike outside, you can expect to have a box similar in dimensions to a travel box--and those are quite large. I flew with one of my bikes once, years ago, so I bought a Performance Bike box; it was plastic but still quite heavy, and around 1' x 3' x 4'.
Found it: it was an Xport cargo box.
(Not my photos).
That's about as small as you'll be able to go without buying a folding bike; both wheels had to come off to fit in that box. May be a good starting point to think about designs. With what material are you hoping to build the box?