In many locales, non -residents are restricted from buying ammo.
Personally, OP, I would contact some good guides in the states you want to hunt. (It would have really benefited you to have made the SCI convention in January as that is when the guides from all over the world book hunts)
There are forms to import your firearm. I have friends who are English and own a home here. They have to provide proof of their need to bring their guns in (an invite to a shoot or a hunt works nicely).
With all of the hoopla going on here about guns in general, I would allow a longer period of time for the processing of the paperwork. For what you are intending to hunt, Montana would be high on my list - the state folks can provide data about licenses and fees and should have a list of excellent guides