I lived on Oahu for 27 years, born and raised. In general hunting and gun ownership is not part of the local upbringing. Most are indifferent to it. The majority of "hunters" do archery or spear wild pig since that's a delicacy there. Traveling with a firearm is not that big of a deal. Lock in case, declare it at the airport, goes in checked luggage. Call the local PD of the island you are planning to hunt. They will direct you if you need to register or not, it may depend on your length of stay.
Locals are generally wary of mainlanders, esp on the smaller islands. They know where everything is and they dont feel the need to mark locations because then everyone finds out and crowds in. Just the way it is there.
Oahu is the urban island and has the most shopping and restaurants. Must see: historic North Shore see the big waves in the winter, Pearl Harbor USS Arizona and Missouri is a must see. Polynesian Cultural Center is awesome too. Everything else is going to be crowded, concrete and traffic. Waikiki is a great place to buy China knock offs and see the police in action!
Big Island is the most diverse landscape wise from desert black lava to lush rain forests and the snow cap peak of Mauna Kea. The volcano park is a must see along with the black sand beaches. Lots of waterfall hikes, ATV, horseback tours. Good Rio Grande turkey hunting there but that would be in the spring.
Maui has the nicest beaches and ocean. Maui is the place to dive. There's some gorgeous day drives, the road to Hana and Haleakalā are amazing. Maui Ocean Center is a must see, esp for the kids. They have dolphins and turtles there. There's a ton of different tour adventures too.
Lanai is where the hunters go for axis and sheep. The island is small compared to the rest. Mostly private. Its always been the place to get away from people and relax on the beach or play golf. There's some ATV and boat tours but everything is extra pricey, esp the resorts. I remember there was a restaurant that served deer meat they got from the hunters. Really not much to do for extended stay with the family.
Personally, I would spend the bulk of my vacation on one of the larger islands (Big Island, Maui or Oahu) doing family stuff and then hop an inter-island flight to Lanai for a few days for the hunt. In Jan the water can be rough in certain areas. Just keep that in mind for diving and boat tours. The Humpback whales will migrate there during that time and you can book boat tours to see them. On rare occasions I would see them from shore.