I don't know how big your kart track is, but the ones I have been to are way too small for something as big as a 600. Your question is kind of hard to answer. A good deal on an already track-prepped 250 or 300 would be a good option in that even if you don't keep it very long you could sell it and not lose money on it. A supermoto would be another good option, but more expensive. You kind of need to assign a dollar value on how much you want to mitigate the risk of damaging your bike. You can get track parts for your bike, but that stuff starts to add up if you don't otherwise want them, and if you're swapping back and forth between street and track setup, that can become a PITA.
300cc if your track is really short and slow. But the problem with a 300/250 is they only look the part. They have no design features that are attributable to track riding/ racing. You will have to do a lot to one to get it to handle decently. Then you will have to relearn to ride the same track once you go back to the r6 after being bored on a 300 as the shift points will be different, the racing line will be different, braking markers, etc. etc. My advice would be to start on the r6, as it is the perfect track toy. Invest in good sticky tires (rs10 or stickier) and take it slow. Maybe even get tire warmers to avoid wrecking on the first or second lap out due to cold tires (which is probably to most common crash among track noobs). Learn the race line, ask questions, stay in the novice group until you are ready to speed it up. Try to find a track/track day organization (like sportbiketracktime) that provides training for all levels of riders. Also it is wise to pick a track that has lots of run off room. Just about every track day I overshoot at least one corner and run off at high speed. It is no big deal, just let the bike coast and don't hit the brakes until the bike has slowed down a bunch. If your local track has no run off room things can get dicey.
BTW i had a ninja 300 for track days and it sucked! I spent about $1500 on suspension, brake lines, brake pads, higher rear sets, etc. and it was simply terrible. Unless you are going to race in the 300 class, skip it.
I am coming off a 2016 ZX10R that was track only, onto an R6 because my local track is really a perfect 600cc track and the 1000cc was over kill.
I disagree with this. Yea, you will have to ride the bike differently, but the little bikes are not terrible. I have a blast on my 300, and up until now I have been on stock suspension! Even big fast Willow Springs was more fun than I thought it would be on the 300. You learn to ride around the weaknesses, and you don't have to throw a bunch of money at the forks to make them much better. And everything you learn on it still applies to a bigger bike.
Just about every track day I overshoot at least one corner and run off at high speed.
Good God, man... you are riding over your head! Seriously, if you are running off track with any regularity you need to step back and evaluate what you're doing before you get hurt. No wonder you didn't like the 300.