As you noted, William, hickory is extremely tough. While I can't tell you the minimum diameter that will meet your criteria in this stool, I can suggest that it will likely be smaller than mahogany or commonly available domestic species. For a while, I was making Japanese fighting sticks, or "jo sticks," with a friend of mine who has a black belt in aikido. In Japan, these are traditionally made of a species of red oak, but my friend wanted something tougher than oak, because of the failure mode and possible danger to the fighters. While oak and many other species often actually break in two upon impact, hickory will flex to the breaking point and then bend over in a tangle of splinters, without separating. Our sticks were 1" in diameter, and, anecdotally, I couldn't imagine a 250-lb person breaking them if they were used in your camp stool. My hunch is that they would flex before they broke. Sadly, I can't guarantee this, but you might want to try the 1" diameter and do whatever torture testing you need to assure yourself.
Another note, typically there is a safety factor built into anything engineered for human support, e.g., chairs, ladders, etc.. This may vary a lot depending on the margin of safety you're after, but it is always greater than 1.0x the design load. I always over-designed things, and I never had anything fail in normal use. You might want to do the same with these stools.
Hope this helps.