Well, a TaG kart would best fit most of your criteria, except for the intitial cost. It would be low maintainance (especially the Rotax), and it would be a kart that you could race either the senior or junior class with. With the size discrepency, you may want to have a couple different seats so the smaller people driving it don’t get beat up too badly. Believe me, you can get pretty banged up if a seat is too big for you.
As far as something closer to your $2k number, you’re pretty much looking at one of the four stroke engines. I’m honestly not too versed on the engines currently on the market, so I don’t know about reliability/running costs on them (though I think most still require more maintainance than the Rotax… could be wrong). Also, I don’t think there is an engine in that category that both you and your kids could compete with, but once again, could be wrong.