I'm a little confused. Do I understand there was a clerical error in typing the VIN number when the title was issued for this trailer? The VIN on the title links to another person and, presumably, a different vehicle. You verified through the highway patrol that this trailer is not the one belonging to the other person. The titling agency has acknowledged that and now you are looking for the correct VIN number on the trailer itself. Is that close?
Stamped into the tongue somewhere is the usual place. Not sure when that became SOP, but that's where I'd start looking. It may be obscured by layers of
paint. Perhaps a Burro owner will chime in with better information...
Too late in your case, but it bears mentioning for the record... Best to check the VIN on the title against the trailer itself at the time of purchase and transfer title immediately after purchase (certainly before putting time and money into a renovation). As you have discovered, it's not always easy to locate on an older trailer, and the oldest vintage units did not get VIN numbers from the manufacturer, just a serial number. Hindsight being 20/20, of course...
Best wishes getting it sorted out!