You pretty much described a dog and then said no dogs, lol. I think a ferret is a good alternative, they can be bitey but I guess so can anything.
Ime, even the friendliest cats do not tolerate the kind of play toddlers want to give and a cat bite can be a lot worse than a dog bite. My cat is a social butterfly but the thing is that its always on HIS terms. He wants cuddles when he does and just forget it if you want love during his afternoon nap (from 10am to 4pm btw lol).
There is a lot of variety in the different breeds of dogs, are you suuuurre none of them will do?