I wouldn't bother with a center dowel regardless. Unless it's totally and thoroughly anchored in the base board that the cake is on it can still tilt and take the tiers with it if the cake isn't handled right.
If I let people pick up a tiered cake (even two tiers) it would be with me carrying it out to the car and making sure it was on a flat surface and secure for the road. I've given clients carpet padding before to make sure things didn't slide around. Just for future reference...That way they're responsible when they drive away, but you can be sure that the cake isn't going to be held crookedly on a lap.
I think there's probably a difference between us holding the cake on our laps and the client doing it. They could have it on their lap and still have it be tilted, which can make it fall over. If the size of the box wasn't the same as the base board that the cake was on it could still slide around in the box, which wouldn't help.
I don't even know what I'd do in this case...Was the cake cold when the client picked it up? How long was the drive? I'd say that if you were clear that the cake had to go on the floor and she didn't put it there then it's her fault, but only if you're sure it was secure in the box and that kind of thing.