We hadn't thought of this, but it would be a good idea to have a hidden ID mark, nowhere obvious, so if your trailer is stolen and recovered you have proof it's yours. The thieves wouldn't know where it was so couldn't scrub it off.
You can also use an EID - I have one on my motorbike, injected into the seat. There should be somewhere on your trailer you could secrete one - inside the jockey wheel maybe, or inside the tubing?
We have our smallholding name plus the breed on the side, but it's on removable panels. You could paint or stencil on your flock name, or your own name, plus breed.