Only thing I could find for Louisiana on the computer is the same as the Florida regulations that I have to go by. If a motor of any type is on the boat then it has to be registered.
Here is what they had listed..............................
BOAT REGISTRATION:
All boats with motors, including electric trolling motors, operating on the waters of the state, must be registered and numbered. Boat registration applications are available from most boat dealerships, and on the LDWF web site, from any of the Wildlife and Fisheries District Offices or from Wildlife and Fisheries, Boat Registration, P.O. Box 14796, Baton Rouge, LA 70898, phone number 225-765-2898.
After completing the application, mail it with your check or money order to the above address and allow 6 to 8 weeks for processing.
Upon receipt of your application and all other required documentation, the Department will issue you a certificate of registration, stating the number assigned to the boat, two decals, and an expiration date of three years from the date of issuance.
Your boat registration certificate must be on board the vessel at all times. Keep it in some type of waterproof container in a safe place where it can be readily found by the owner.
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Personally I would locate a Game Warden and ask them since they are the ones that would enforce the law about that. Florida has tried to make it so a person has to register a boat powered with only paddles to be registered but everytime it is shot down , some states have made it a law that any boat has to be registered.