I've done Kruger a few times, each time with different kit. My previous trip I only had a 70-200 with a 1.4extender and I really suffered on length for a lot of the shots. I now have a 150-600 sigma and for me it's ideal for the large open expanses that are very common at that park.
That being said, I think you'd be content the majority of the time with 400mm on a crop body so with your budget in mind my advice would be a 2nd hand 70D (~$500), a 2nd hand 70-200 f2.8 (non IS) (~$800), a Sigma 2x extender ($300) and possibly a 2nd hand 18-135mm or using the kit lens for shorter stuff which you should be able to get included in a 2nd hand 70D package. Throw in a spare battery and you'll probably be touching the $2000 mark.
The advantage to this setup is you'll get good range while still maintaining good performance and can remove the 2x tele if you need to drop back on reach and bathe in the glory of what is an inspiring lens which would serve you very well after this trip in whichever photography endeavours you take. The lack of IS shouldn't be a major issue as there is generally very good light the majority of the time. If it does get dark you can also run without the tele and have a very fast lens which will be as capable as any in lower light conditions.
Trips like these usually don't come often so if that's the case for you it's better to spend closer to your limit to ensure you are equipped with decent gear.
As mentioned above though, if you have no real intentions to do photography afterwards or it's not really a big interest then there's lots of cheaper and easier options.