I don't have one but the local fun track (Speedys in Houston) has
Shaller karts and sells them every once in a while and I almost bought one w/ a gummed carb for $600. They seem very well built.
The razor looks like a good option and I looked into but I went w/ a
Razor Dirt Quad instead and my kids love it. It gets up to 10 mph so good for kids 5-8 or so but doesn't have the excitement of a gas engine behind you.
Segway has a g
o kart kit that uses a hoverboard type thing to power it so you can ride it separately if you want to.
Bet his worrisome family would flip if you got him a dirtbike or real 4 wheeler. I've gotten my daughter an XR50 & XR80, both for well under $1,000.
Either way, hope your kids have fun be safe.