If you are worried about it you could just type up a simple contract stating that you (the hauler) are not responsible for any injury's or possible death to the animals. Likewise they are not responsible for any damage their animals could cause to your property (trailer) and everyone signs it.
^ This one.
Whatever course you decide upon, there is always someone to tell you that you are wrong. There are always difficulties arising which tempt you to believe that your critics are right. To map out a course of action and follow it to an end requires courage.