If the police run a background check on you before you are issued the permit, you are exempt from the GFSZA. I would recommend that someone get's a background check from the police or AG (whoever does it in NH) and bring that to your licensing authority (if it's your town selectman or something) and include it in your application for a license, that way, according to the exemptions laid out in 18 USC 922(q), you would be exempt. I would imagine most PD's run a background check on you before giving you a license, I know the state police do for non-resident licenses.I think what it boils down to, is that it's just not an issue in VT. People from VT aren't here participating, because they mostly just don't have any issues: they just carry, and that is that.
I keep up with the news from northern New England, and I can only recall one case, from about 4-5 years ago, where someone was charged with having a gun on a school campus in VT. I don't remember the particulars, other than that he was being actively pursued by LE for another crime, and jumped the fence to an elementary school and ran into (and I think briefly hid in) the school building.
My "dog in this fight" is that I'm moving to NH, and depending on which part of Coös County I land in, I'll either be just a few miles, or just a few yards, from VT (or ME). NH is pretty skinny that far north, so it pays to know all three states' laws.
NH, by the way, also doesn't have an exemption to the FGFSZA, because licensees don't undergo a background check. NH has no laws against carrying on campus or inside school buildings. How does NH "get around it"? Nobody cares, just the same as VT.
Nobody has ever been charged under the federal law unless they were also doing something that brought the feds into it, like dealing drugs in a school zone. The odds of a local or state LEO referring a gun possession case to the feds, when it doesn't violate state laws and there are no other violations in play, are simply miniscule. Could they? Sure. Would the feds pursue it? Possibly, but not likely.
If the feds pursued it and won a conviction, it would open up a new challenge to the post-Lopez modification of GFSZA. Absent other federal violations, they'd probably lose, and they don't want to lose that carrot when it comes to carrot-and-stick plea bargain negotiations. GFSZA charges are always an add-on charge, something they settle for when the defendant agrees to a plea bargain.
Secondly, the background check thing in itself may not be an issue, Alabama does not require a background check to get a carry license, but the 11th circuit ruled that an Alabama permit will exempt you from the GFSZA.
I would argue realistically that the only way a New Hampshire license may not be valid for an exemption would be if it's issued by the mayor or town selectman, and they do not run a background check on you at all, and you do not include a background check in your application.