If the employee at town hall is the one that messed up and set your GVW at 13K pounds, you can go back to that person and have them fix the issue. Having the title and registration updated to reflect the proper weight shouldn't cost you anything if this was their error - BUT the registration itself WILL cost more, since we are effectively taxed on the vehicle weight at registration time. Empty weight of a deuce without a winch is right around 13K pounds, so technically you are overweight with a full fuel tank and a cup of coffee. Add a payload, and some zealot with a set of scales could cause you a world of grief to sort out. That said, in all the years I've owned a deuce I've never received more than a wave from law enforcement in NH. YMMV. Myself, I run regular plates and get charged for full weight (23,535 pounds if memory serves), just so they won't have anything to complain about.
I'm with you on liability insurance - you could lose everything without it, just because some knucklehead did something wrong and ended up underneath you. Wouldn't roll out of the driveway without insurance.