Been doing this for 40 years or so.
Not getting any easier, especially under the present conditions. The air freight price has gone through the roof of late. I have a box of trophies ready to ship to Chicago and then onwards to Michigan. Just the one way freight to Chicago is double what it used to be!!
First of all everything that is shipped via air has to be in a wooden box with air craft grade timber (present box I have is £150 made). The handling agent this end needs to make sure all the paperwork is correct. Scientific names and common names of the trophies, where shot, and a letter from the Outfitter or guide stating that they were all taken in accordance with British game laws. The shipment MUST go through a designated port of entry in the States, with a Fish and Wildlife facility to be checked. This is where you will need a good company handling this in the States.
There are not a great many companies handling such goods in this country, and who actually know what they are doing. Many will not deal with one of individual boxes. I use a company in Kent, who in general have been pretty good. You can PM me and I will give you their details.
You may also want to consider making sure that if two clients have their trophies in one box that it only goes under ONE name and forwarding address. The Fish and Wildlife in the States are getting twitchy about multiple trophies in one box.