You could also do a cheap and easy swing arm rear suspension also.
Easy is correct. You can even take a hardtail kart and literally cut it in two, and just put it back together with hinges. Voila, you have a swingarm. Just add shocks. Cable the brakes
and the throttle.
There's another way to do it that leaves your motor stationary. You mount the motor rigid above the swingarm, and then run the primary chain from the clutch or CVT or whatever the motor output sprocket turns out to be to a jackshaft. You run the secondary chain from the jackshaft to the rear axle. The trick is to hinge the swingarm in the
exact same plane as the jackshaft sits. That way the secondary chain tension will never change as the swingarm goes up and down, while the motor sits nice and still above a kart with suspension under it.
Pics below show one of those, with some special geometry added to the swingarm. The shocks are 17" nitrogen-filled adjustable coilovers, mounted in a cantilevered geometry that actually reverses the action of the springs. They act with diminishing resistance when compressed, instead of the resistance increasing. No need to get that fancy, the point is to show the stationary motor above a swingarm with a ridiculous amount of travel.
This particular kart got a 44" axle for the swingarm/jackshaft, and
after the bearings were mounted, the axle got cut into 3 pieces so that only the jackshaft section would spin with the chain, while the 2 swingarm sections just go back and forth.