Undercooked beans in chili?
It looks to me like you have a couple of things going on here:


You are worried about the safety of undercooked beans
You want your beans to be palatable


To address the safety issue, I think we'd need to know more about the kind of beans you're using. Kidney beans are the ones most well known to be toxic if they are undercooked. Other beans, say, great northerns, or pintos, wouldn't have the same problem.

Then again, you still want a chili that is edible.

You certainly can cook dried beans, but there are a few factors getting in your way. 


They do take a while. I cook beans fairly often in our house. Even after an overnight soak, they still often take (depending on the kind of bean) 2 1/2 hours at a low simmer. 
Then you have the acid issue. Most chili is tomato-based, and even those that aren't usually have some other sort of acid. I'm going to go out on a limb and guess that yours does. Acid does inhibit the softening of beans, and this can work for you or against you. Many (non-texan) chili cooks find that the acid keeps the pre-cooked beans from turning to mush. On the other hand, the amount of tomatoes in most chilis means that you could boil for weeks, and those beans will probably remain hard.
Lastly, I'm guessing that you don't cook with dried beans too often. I don't mean that to be an insult--I've actually done the exact same thing as you, and found myself coping with a pot of chili full of hard beans that I owed to a football party. The reason that I bring this up is that, believe it or not, dried beans actually go stale. If they're old, they may never soften.


As I mentioned, I did have this same thing happen to a pot of my chili. If memory serves, I sat there fishing out each bean for probably longer than it was worth.