The bottom line is no matter how many times you "make yourself clear" that guy from Australia CANT buy a firearm in Hawaii. 
They can, but not within a 11 day visit. The citizenship requirement is part of HPD's process, not the law. HRS 134-2, has provisions for non citizens that require additional background checks. That being said, the entire process is just too much of a hassle for someone here on vacation.
avalasia: It can be done but won't be easy or even worth it. Unless the "right" people are working the window, expect to spend an entire day at HPD if not more (multiple visits) just to get the application. After that, the additional background checks will add to your wait. The way I see it, calculate the cost of extending your stay to 1 to 2 months just for a permit to acquire (with no guarantee it'll be granted). By the time you add it all up you'd be better off just paying the higher price at home. Also, I'm not 100% sure but they may have restrictions for transporting firearms on international flights. BUT I'm a glass half full kind of guy, so as an alternative to the whole permitting process, try emailing dealers in Hawaii before your visit. If there is a way for you to pay for it in person, then have it shipped/transferred, they'll know.
As for the states you can buy long guns in, I believe it's pretty much any state without permit to purchase stuff. Hell, you can do it in Chicago with no FOID card as of a few days ago. (new ruling)
I'm trying to figure out what to do while I'm on the big island. I'll have to get a permit, but my perminant physical is in Alaska. My temp is on BH, but I have no paperwork to support it. *shrug* will probably just work on my ar and buy stuff when I get home and resume FFL work.
As always, state and county laws apply. California is a no-go (big surprise). But I can confirm those listed by Tom_G from personal experience. And add Nevada (except Clark County) & Florida. Personally, I prefer gun shows over pawn shops but I got a great deal on a Benelli O/U last October in Killeen. Just showed my Hawaii drivers license, paid, and walked out the door with it. For handguns, not all FFL's in the mainland will hassle you about shipping it out of state. Most will allow you to take possession after their state mandated waiting period and let you handle shipping it yourself.
Have you considered using your work address for the permit application?