I may be wrong but I believe that NH requires you to have a LTC in your state of residence in order to issue a non-resident LTC. This is because they don't actually do any checks, they rely on the license copy that you send them in their non-resident application.
Your best bet IMO would be just to get a Massachusetts FID and forget completely about CCW in NH until you're 21 because you can't handguns til you're 21 to begin with. Remember that in NH and most other states no license is required for mere possession, transportation, and use of firearms.