Since no one else has taken a shot at this I’ll throw in my “any feedback is appreciated”. ;)
While I don’t have the direct mechanical experience, I recently drove some at an indoor track outside of LA last month and coming from a sprint kart I remember thinking how poor the handling was (significant push and very sensitive to snap oversteer). I attribute a lot of this due to a massive lack of balance in weight. Of the 600 lbs, I think 599 of it is towards the rear due to location of the drivetrain and batteries. Here’s my thoughts:
1) First thing I would do if it were allowed would be to get rid of the rental kart tires and throw some race tires on it. My suggestion would be to try out some MG Reds or Yellows. The only caveat is it’s a crap shoot as to how race tires would react to an indoor surface?
2) Is there any way to weigh the kart and find out what the actual weight distribution is? My gut tells me it would be very rear heavy and if this is truly the case I would look for any manner to shift the weight towards the front (or try adding some weight to help the balance). For reference a sprint kart typically is around 40% front to 60% rear weight bias.
3) In terms of front end setup, I would suggest widening the front, add caster, and a little toe out. This is what I would do with my sprint kart but keep in mind that this may just result in more snap oversteer due my belief of a massive weight imbalance.
4) Is there a way to raise the front ride height to increase the leverage resulting in more front grip?
Please note that these are things that I would do to my sprint kart and there may not be a directly correlation with the kart you are working on. I just wanted to throw some ideas at you that might help.