I would lock it down as you suggested. Marking the trailer is OK but may not help much. I had a friend in my RC flying club had one of his utility trailers stolen even with Usual padlock through the latch and a lock on the tongue and very distinctive trailer markings with airplane decals etc. He reported the theft in less than 12 hrs but has never seen the trailer since along with several thousand dollars worth of distinctive model airplanes.
About the only thing that can be done is make it as difficult as possible and hope they pass your trailer for another one.