I know this is an old topic, but I thought I'd answer anyway.
There is a lot of NFA stuff in NH. It is a very NFA friendly state.
The idea of consenting to a search by the BATF because you own a transferable NFA weapon is a wive's tale. Once the transfer is complete, it is yours, like any other firearm. You'd have to check into the legality in other states that you wish to take it to, but in NH no further registration is necessarily.
What are you planning to buy?