In my opinion it would'nt stress out a rabbit to be carried from house to house in a basket , I often took the late Nash-eoke and his shop bought and now ageing sister Toky-Ohki from Barbara's to Grandpa Rex's just around the corner from where they were kept.
However a short distance nearby in the U.S is a lot longer than a short distance in the U.K , so I will leave it to you discretion.
Too keep =^-^= Nash-eoke and = ^ _ ^
= Toky-Ohki quiet we always used to put them on top of an office swivel chair, it was high enough off the ground to stop them jumping off , but the arm rests gave the rabbits them them feeling that they were contained in a burrow and secure so they eventualy settled down so far that even their noses failed to move almost in the land of nod.
=^o^= Schneeball on the other hand is a Chinchilla blue and thus his breed makes him bold and brave , so he did jump off every now and again.
My most recent Rabbit buck =^o^= Schneeball has been carried in a basket , hopped around the corner on a harness and has frequantly been carried in a basket behind my bicycle in a coolstop trailer.
So I dont suppose car travel will be much of a problem , it probably depends from rabbit to rabbit.