Satrider, I'm sure you know this, but I think it needs to be said again. Some horses don't like being ridden bitless, it isn't always a matter of needing reinforcement. I ride Soda in a double jointed snaffle (much like what CM posted). It's an extremely mild bit and I rarely need more than a twitch of my fingers to direct him. But he hates bitless bridles (Dr. Cook in particular). I think it has something to do with the pressure around his face, but I could be wrong.
I'm sorry for bring it up again, but I don't like the conception that if a horse isn't ridden bitless it's because they "need" something stronger. For him it wasn't a matter of strength, more of how the pressure was applied.