Henry,
You could call TS Racing in Bushnell, Florida, since they are listed as a dealer for that clutch. They probably have the information you need.
If you’re not using this kart on a track, you can expect to generate more heat in the clutch because it’s spending more time slipping. Try lowering the engagement rpm to keep the clutch locked up more. Less slippage should lower the temp. How much? Couldn’t tell you. I’ve never checked the temperature of a clutch when it comes off the track. It’s hot and I respect that.
Good luck and enjoy the time with your kids.

You could put a 12t on the clutch and/or a bigger rear gear and it should be set to the lowest lock up for yard use. 250 after 20 min don’t sound too bad.