I've seen where people have put a trailer on a class 6. they're kinda odd but functional. Look who's talking, right?
Those toy haulers are pretty tall. how tall would that be on a class 6.7.8? Mine is 12.5 feet and that gives me problems at times. The wheels on that trailer are probably 15" and a truck will have at least 22.5" wheels. That frame is much higher.
Just FYI, I bought a truck with a lift gate and we can load bicycles, motorcycles, even a lawn tractor once or twice.