I mean no disrespect, but are you certain of that? I've looked into the NH law's as well, and please correct me if I am wrong, but from what I've gathered the only time you need a permit, so long as you are 18 years old, is if you want to carry concealed. You don't need a permit to posses a rifle, shotgun or handgun so long as it is to and from the range or if you carry your handgun unconcealed.
Even so, I already filled out the application for a Non-resident pistol permit a few months ago and I put together some other stuff as well. I'm stilling holding off on sending it in because as you stated, they require you to have a permit in another state, and I don't have my FID card yet. But FID isn't a permit that covers handguns so I don't know if that's acceptable. I'm hoping that they'll understand that I can't legally have a handgun permit here in MA so they'll still issue me the NH non-resident pistol permit. We'll see.
Thanks for the help guys!