He will probably not be too bad.. especially if he has never been used for stud.
Cover all the car interior fabric that he can touch with polythene sheeting... that will prevent smell being transferred!! Put the rubber matting on the floor area.. this well hold the polythene in place as well.
It is quite legal to transport this way for 'hobby breeders', because when you get home, stripping out all the covering constitutes 'cleansing the vehicle'.