You have already gotten recommendations for good agents. I would just add that I would suggest that you do use a travel agent recommended here and also using a rifle permit agency.
Overall when you come back you might feel a bit like you had to pay many different people for so many different services. But, a travel agent and a rifle permit company (in my opinion) are well worth it. Even better if you can book the rifle permit agent through the travel agency.
I have (so far) been happy that I can make one fairly short phone call and have domestic, international, and inter-country connecting flights booked. Same phone call, my hotel is booked, my rifle permit is applied for, my rifle permit agency is arranged, my layover hotel is booked, and if I desire, my meet and greet at the airport is arranged as well.
They are human, and I always double check things are correct. Sometimes they aren't. I have had some errors on a rifle permit, and I have had dates wrong on booked flights. So far, I have just called and pointed out the error and they fixed them promptly.
There are times where I wouldn't recommend it because it isn't necessary. Like if you are traveling to a country to hunt, but aren't bringing guns, and don't have many or any short connecting flights in other countries. Then book your own and save the money.
Guns complicate it, layovers at hotels with guns complicate it, and connecting flights in other countries with small airlines while bringing guns complicate it. You can do it all on your own, if you want. I don't have an interest in doing it.