Are the people at your DMV trying to look the trailer up as a car or motorhome? 17 digit VIN numbers didn't become standard on cars until the early 80's.
A quick check at Airstream and the Vintage Photo Archives show that your VIN will be 10 digits. How did you come up with 16? Did you add numbers from the placard by the door?
I'd bring photos of the build plate, tongue stamping if you have one, and ask to talk to a supervisor.
AirForums # 2806
WBCCI / VAC # 6411
Not All Who Wander Are Lost.
Avid supporter of trailing edge technology.