Firestorm is correct here, the requirement is the "intent to make a home" in the state, which means your residency can't be purely transient. Staying in a hotel once a year doesn't work because a hotel is by definition transient. Purchasing interest in a property, even for one week a year, has a level of permanency. Actually, it still would be legal to purchase while living in a hotel if you have a bona fide intention to make a home in that state. Say, if you're staying in a hotel because your job transferred you, and you're looking for an apartment.
The caveats here are, 1) you can only buy while you're actually residing in the state, which in this case would be only once a year during your timeshare week, and 2) you have to convince the FFL to accept your proof of residency, which really means you need a NH ID in this case. They'll mail the ID to your timeshare address and I'm not sure how receiving mail works in that scenario.
That said, using a timeshare for residency is probably not a great idea, it was just interesting to discuss