I did a 6 week trip across the States on my GS the other year. I used James Cargo, works out at £1950 to fly in and boat out. We flew into Las Vegas and boated out of New York. Flying in is definitely the way to go. Less time without the bike before the trip, plus it's more reliable. Getting it on a plane is as easy as dropping it off at the depot of your chosen air freighter. They pallet it, box it and deal with all the paperwork this end. It even goes with a bit of fuel in, keys in the ignition and all your gear, including helmets, in the box with the bike. Easy.
Getting the bike out of cargo is a bit harder at the other end, but patience is key. Go to cargo lounge, get paperwork, take paperwork to customs, explain what you're doing. Return to cargo with paperwork and you're free to go. Usually a $50 fee at the cargo lounge. Las Vegas works well for those fresh off the plane as two miles from the airport you're in the desert. No interstates to endure.
Coming back you're a bit more on your own as James Cargo et al just rely on third party agents in the States. Leave the bike with them, deal with some paperwork and hop on a plane home. Six weeks later pick your bike up from wherever it comes into port.
Before the trip you need to apply for an environmental protection agency exclusion letter which are easy and free to get. Your shipper will point you in the right direction. No need for a Carnet.
Insurance I've always used Motorcycle Express, who do do shipping but also offer full insurance on the bike, plus breakdown cover. Worked out about £300 for 2 months I think. Some people use Progressive which is cheaper but you do have to fudge that you have an American address. How valid the insurance is should you need to claim I'm not sure, but then not heard of any complaints.
I'd definitely recommend taking your own bike if the figures stack up. Anything under 3 weeks it's cheaper and less hassle to rent, but anything longer and you're saving money the longer you're there. A GS 1200 from Las Vegas BMW was about $250 a day last time I checked. Some people say buy one out there, which is fine if you can afford to be in ownership of two bikes, which I can't. And having a UK plate in the US doesn't half get the conversation started.