
What world are you living in where a 600cc street bike weighs 350lbs? Maybe 370 dry but 350lbs? Wet? No fucking way.
Still, a Honda Element should be plenty capable of hauling a 400+ lb bike provided you could fit it inside and strap it down securely.
I'd probably go with a small open trailer though. Much easier to load and you can strap it down better. A friend of mine has one he bought at Harbor Freight for around $200. He made some ramps for it and mounted a device to hold the front wheel. He tows it behind his Hyundai SUV.