Don’t know if you’ve ever watched the movie Faster about Moto GP (it’s on Netflix), but toward the end there’s a neat bit about how 3-time world champion Wayne Rainey, whose legs were paralyzed in an accident, switched to racing karts and really enjoyed it. Like you, he had to modify the karts to operate with hand controls only.
Italian motors sells a hand-brake with the master cyl mounted on near the steering hub. It’s for operating supplemental front brakes on a TaG kart but I don’t see any reason it couldn’t be adapted the run a rear brake too (most TaG karts don’t run front brakes), just by re-routing the brake lines. You’re half way there and the hand throttle should be a breeze.