Do you already have the riding mower in hand or no? The only time a mower would ever have a centrifugal clutch is if it's a race mower, either FXS/XT (Factory Expiermental Single or Twin) or any other class of tractor. Some do run the belt clutch, however if they are just bombing around the yard or something or having friends/family ride it, they have to swap out the belt quite often.
It would not be hard at all to make a kart with a vertical shaft engine, but it will be expensive. To keep it compact you would need to build it in a front engine, not unlike a lawn tractor. You'll need to make a chain tensioner and from the clutch to a right angle gearbox, (I'd recommend at least a 2:1, 2 revolutions in 1 out), on the input side of the RAGB, 25 tooth, and 10 on the output. On the axle, have a 60 tooth sprocket. With all that, the rear can be either fully independent suspension or a swingarm like you find on a utility ATV or race quad. With that much gear reduction, I believe you will not burn out the clutch, however you will not be going very fast.
If you do not have an engine, i would strongly suggest getting a horizontal shaft engine. Briggs, Predator, Kholer, Honda, Champion, Rato, etc as it will be a bit cheaper than buying a right angle gearbox to take a vertical engine to a horizontal. As well as not having to worry so much about chain alignment in that orientation. Its takes a bit of imagination.