I'll leave the interpretation of law to many who have posted who are very credible. I will take the point that as said it is legal to carry an antique. As the law states it is legal, that is agreed on. So yes you can carry an antique firearm in NH as the law describes.
So now the question becomes, because of factors beyond the law " the understanding that an LEO may have of that law" is it worth doing.
That comes down to reason for carry, cost in time and money. Being that you are likely to be arrested if caught = Time. Need bail and an attorney = $$$. Then a hearing or trial which = more time and money.
Because the LEO probably is rarely put in a situation to have to make this distinction between lawful carry of an antique or unlawful carry of a firearm. In this case my guess it that they are probably going to fault on the unlawful side, arrest you and let the DA or someone else figure out the situation. Costs the LEO nothing, but can cost you a lot.
So I guess, yes you can do it, but you can also run into a situation that results in aggrivation, time and monetary loss. Not fair, may not be right, but seems to be the likely outcome.
So the question becomes is it worth it to you. If so, have at it. If not, don't do it. Yup it sucks that this is the way it is, but it is. As long as all guns and people who carry guns are considered "bad" by a large part of society and the laws are so complex and unclear that the "average guy" can't make sense of them, this situation is what it is.
So if there is a highly likely hood of a threat to you and/or your family. probably worth it. If it is being done to bring light to the point that it is legal, doing it, may worth it. These are all personal preferances.
There are many other similar laws, interpretation of laws and how the law will be enforced. An example I can think of was, early in my carreer I was driving an ambulance, with lights and sirens activated. My partner was in the patient compartment treating a patient with chest pain. We traveled safely above the speed limit and stopped at a red light prior to passing through the intersection. We never came close to causing an accident and the day was sunny with dry roads. I operated above the speed limit and proceeded through a red light "with due caution/regard" as the law allowed for an emergency vehicle.
Well on arrival at the hospital an LEO who followed, he asked me why I drove with my light and siren and then tickets me for marked lanes violation, speeding and running a red light. I pleaded my case and stated the allowances given under law to an emergency vehicle. But the LEO disagreed stating in his "opinion" our patient's chest pain was not "life threatening" and applied his interpretation of the law.
I fought the tickets with the help of my employer. So we went to court, the court agreed that a "life threat" did exist, that we the "medical professionals" make the determination of "life threat", not the LEO and all counts were thrown out. The person hearing the case said I was acting lawfully. Cost my employer a lot of money on attorney etc.
Now if the patient had a non-life threatening situation or we were simply tranferring a patient between hospitals as ambulances do every day, the LEO would have been correct as an "emergency" did not exist. So in these cases the ambulance must travel in traffic, with posted speeds and obey all controls.
Before I get decended upon my example is just to factually demonstrate how an LEO may not know every aspect of all laws. In this case the LEO was new and inexperienced. But his powers are no different than those of a veteran LEO. He made the wrong interpretation and took action against me. In the end I was proven right and he wrong, but the only ones out money and time were my employer and I.
I guess that's why they say "this is why courts exist." So the question is really, is it worth it? Not is it legal and can you do it?
My 2 cents.