I have to say I don't really know how double-jointed bits work, to be honest, so I'm not sure about the eggbut bit you're describing. Whatever you start them on, it should be relatively gentle. Dulcify mentioned the thicker bars, and I agree that it's best to start them out with thicker rather than really thin bars.
It was so long ago that I last used the key thing, I'm not sure if I remember things accurately. I think the horses did play with the bit quite a bit, but once I switched to a different bit, it was fine. Neither of my horses ever got a tongue over the bit, but maybe I was lucky.
Now I just use an open snaffle, single joint, with fairly thick bars to start. I haven't had problems with reins moving around on the bit or anything. After a few rides I move to a slightly thinner bit. I have one three-year-old in a single-jointed copper D-ring now, and she's responding well to that (she started out on an open snaffle, but didn't yield as nicely as I like).
I don't generally use full-cheek snaffles because of the way my reins are set up. I guess they're probably fine, though I do wonder if they would put a little pressure on a horse's lips (not that I'm super worried about that after the horse has a few rides on it - I believe the D ring does the same).
I guess I'd say just go with something relatively gentle to start out. Good luck!