I think we're overthinking the cooling thing, and missing the obvious-
***THERMOSTAT***
Surely to god a thermostatic system can be employed here. I'm not familiar with the inner workings of a jetski engine here, but I find it very, VERY difficult to believe that a modern jetski engine simply inhales cold water and circulating it once, exhales. How could the engine possibly reach operating temperature? Running an engine cold all the time is not only inefficient, but downright damaging. So- that leaves two possibilities. One, that the original cooling system uses a thermostat to keep engine temps in check. Two, the system draws in the water, and has a small outlet- allowing the water to circulate around quite a bit without being constantly exchanged with COLD water.
If the former is true, the OP can continue to use the thermostatic system, and as long as the radiator and water pump are up to the task, shouldn't be a problem. If the second system is used, it might take a bit more fiddling to get right, but should still be doable.
Converting to a closed system may take some doing, but it shouldn't be Mission Impossible. Install a water temp gauge as soon as possible, and keep a really close eye in it until all the bugs are worked out.
You'd also be wise to adapt in a radiator with a pressure release cap, so that if the numbers get a bit high, you don't blow a hose. A small recovery tank would be a good plan, too.
By the way- I'm going to play Devil's Advocate for a minute here, regarding the whole boat VS car thing here, regarding HP, drag, etc.
First, HP is HP, torque is torque, and if the jetski engine develops, say, 65hp at 7000 rpm, that's PLENTY for a kart- think about it.
Second- efficiency. A propeller (or impeller on a jetski) nearly ALWAYS slips. There is no mechanical lock-up, as there would be between a kart engine and it's wheels. A propeller is like a clutch that is pretty much always slipping. Partially because of this, it takes a lot more hp and RPM to get a boat moving, and keep it moving.
Third- Drag. Until a boat gets up on plane, there is a significant amount of drag involved. Someone above compared pushing a car (or kart) on the ground, verses pushing a boat. I would suggest that pushing the boat is maybe a bit harder in comparison than you think.