Maybe we can take a few tips from the old rabbit breeders of yesteryear...
Many years ago, up until the 1950's, it was possible to send rabbits unaccompanied by train, to shows. The owner would put the rabbit on the train at one end, and it would be met and cared for at the other, by show officials. This was the way for cats, rabbits, guinea pigs, pigeons, poultry and many other animals for at least 50 years.
Part of the secret of the rabbits travelling well - and arriving in good enough condition to compete at a show!! - was the design of the travelling box. It is now getting hard to find people who make these heavy traditional wooden boxes, but they are basically small, dark, enclosed and warm - but well ventilated using a system of vents & baffles, and as strong as iron to protect the rabbit in case of rough handling by railway staff.
Rabbits like the dark, and the feeling of an enclosed space, like a burrow. If they are frightened, they will stay still and cope, rather than panic, and the size of these boxes reflected that. Too big, and a panicking rabbit could hurt itself. Also if there was some kind of accident (or even braking sharply in a car these days) there wasn't much room for the rabbit to be thrown across the box.... like a pea in a biscuit tin! They also tend to be much happier if secure in the dark, rather than being able to see out. If you have a rabbit which doesn't like travelling, I would definitely recommend something dark & snug.
I once knew some people whose French Lop loved to spread himself along the back window ledge of their car. The owners delighted in this, and enjoyed the looks they got from other motorists! He travelled up & down the country to various shows, until one day when the driver had to do an emergency stop. Not only did the poor bunny, flying across the car, almost render the driver unconscious when he hit her on the back of the head, but it was a miracle he didn't go through the windscreen. As it was, he ended up wedged under one of the pedals.... and he travelled in a small enclosed dark box after that, safe and secure, and much less stressed.
There are various recommended sizes for these type of boxes, according to breed/size/weight, and I'll be happy to post the sizes if anyone's interested.