I've shipped my moto (a Honda ST1100) back and forth from Canada to Europe a number of times.
The trick, if you don't want to pay full retail price for the 'cubic weight' of the motorcycle, is to find a charter airline that operates seasonally between your desired origin and destination, and uses wide body aircraft. The charter airlines generally restrict passengers from bringing excess baggage on board, and if the aircraft is a wide body, it will have lots of empty space (cubic space) downstairs.
A motorcycle, no matter how large, weighs virtually nothing in the grand scheme of things - a LD6 container (pictured below) can contain up to 7,000 pounds, if you put a 600 pound motorcycle in it, the weight of the moto is negligible to the air carrier. Often, the empty containers need to be moved around anyway, so if the charter company can pick up a few extra dollars for tossing a moto into an empty ULD that they have to reposition anyway, it's "money for nothing and chicks for free" as far as the air carrier is concerned.
Out of Vancouver, you might want to investigate shipping with Edelweiss, who run a summer service between Vancouver and Zurich.
I have always used Motorcycle Express
, a New York State based company, to make all the arrangements for me. They are not cheap, but you get good value for the money you spend - they arrange the pallet for your moto, book the space, do all the paperwork for the shipment, and let you know exactly where and when to drop it off and where and when to pick it up. They have been doing this for at least 10 years and they are good at it. They can also book you a passenger ticket on the same flight as your moto is scheduled to be on, but be aware that the moto travels on a 'space available' basis, and might get bumped to the next day or even a few days later if the headwinds are strong and the pilot wants to carry extra fuel.
The most attractive prices are for Air Transat flights out of Toronto or Montreal (ideally Montreal) into Paris. I've shipped out of both those Canadian cities without any difficulties. It's always been pretty easy to get the moto out of the freight shed and cleared through customs in Paris, so, I recommend Paris as a shipping destination. I strongly discourage you from shipping into the UK, simply because the import formalities are a real PITA there.
You might want to consider riding across Canada to either Toronto or Montreal, then shipping from there. The cost savings would probably be (I am guessing here) about $1,500, which would pay for a good chunk of the Trans-Canada ride.
Packed up, it looks like this: