Two federal agencies are involved.

The Department of State has jurisdiction over the exportation and temporary importation of firearms into the U.S. To transit the U.S. with a firearm requires that you temporarily import the firearm while you transit, then export the firearm when you leave. The Department of State Office Defense Trade Controls issues those permits. ODTC

ATF has jurusidiction over permanent importation. ATF regulation use to say persons coming from a foreign country to the US for legitmate hunting and organized sporting events could bring their firearms. The ATF regulation did contradict State department regulation, but was simpler and the accpeted procedure at the border.

After 9-11 ATF changed the regulation to require application and issuance of a permit-ATF6 for non U.S.Residents to bring firemarms into the U.S. for sporting events and Hunting. You can get your questions answered about this form at ATF Form 6

Just quickly scanning the FAQ's Q6 it looks like that you need a State Department temporary importation permit, DSP 61.

If I were still in the field doing enforcement work, that's how I would read the regulations also.