I have clients from the USA eastern shore this year, and have dealt with many American clients over more years than I care to remember.
There are several ways to get to Inverness from the USA.
Fly direct to London.............. Most USA flights land at Heathrow, but Gatwick is also used. Take an Easy jet, Flyb, BA to Inverness. All take firearms. RYAN air does not allow weapons, but as fare as I know does not service Inverness.
Fly into Edinburgh ...............its been a few years since I collected clients from this airport, but it might be worth looking into this option, and again taking the train up to Inverness.
Fly direct to Glasgow..............Take the train to Inverness, brings you right into the city centre. Pick up is fairly easy but the entrance on the high street is pretty limited for waiting. Train journey is about 3 hours.
If you are carrying firearms you will need your British Visitors Firearms permit, this must be signed by you and shown to customs on arrival in the U.K. You will also need to show this on departure from the U.K. Who ever you are hunting with should sponsor you and arrange this for you. Cost is now £20 per individual, or £100 for 4 licences and over.
Hope this helps.