There isn't anything special with rabbit carriers, really. If the cage can be placed under the back seat or secured with a seatbelt so that it wouldn't slip around on the ride then there won't be much of an issue with that. If Bullseye can move around free, but not too much (eg good to turn around and maybe do a few hops here or there) then that's fine too. I don't know what type of walls the cage has and if it has and possible effects though.
For a shorter ride, hay n blanket you don't mind being soiled is enough. If it is longer, then you could take a spare blankie and some water with you. No matter the length of the drive, i also like to carry around a small bunch of pellets or vegs for the bun to chomp on at the vet, if they take it. Other than that, evertyhing should be okay!
