I would say you have a couple of options. One option would be to bring a gun from the U.K. to the US. I don't think there is any problem with that as long as it is in the cargo hold. I know there are U.S. rules, but what the international or British rules are, I do not know, you would have to check on that.
A second option would be to buy a gun here after arrival. Perhaps you could arrange with a dealer (prior to purchase) for him to buy it back at some specified amount, when you are ready to return home. I don't know what the rules are regarding selling firearms to foreign nationals are. However, if you stay away from a dealer and buy from a private citizen there is no paperwork involved. It is a private transaction period. There is no license required to own a rifle or shotgun in most jurisdictions. There are hunting license requirements in all jurisdictions. In addition, some wilderness areas have a no firearms requirement. That varies from place to place. You would just have to check into the specific area you are going. Whether it is a state park or a federal park will make a difference. One important thing: years ago none of the states required a hunter safety course. Now they all do. Old guys like me who had been hunting for years and years (and had not shot ourselves, lol) were "grandfathered" in and not required to take the course. Younger people born after a certain date are required to before they can hunt. But you wont be hunting so that shouldn't be a problem.
Another option, and this would be my recommendation, would be to begin to correspond with someone in the area regarding your trip. After you get on a friendly basis, then ask them if they could arrange the use of shotgun (or whatever) for the period of your stay. You might have to put up a deposit -- in case you lost it in a lake or brought it back covered with rust, etc, etc.
If you have shot clays on numerous occasions you must know all the basic fundamentals. How to load it, how to aim it, where not to point it, and of course, to always stand BEHIND the trigger guard!
All of these things are just something to look into and make some decisions. Like I said I wouldn't worry about the bears, but I certainly would take some bear spray as was recommended by others.
In some areas of this country, no type of firearm will raise any eyebrows. In others, a pistol might, but not a rifle or shotgun. A shotgun would be the least apt to cause any undue attention.