It depends on the car you drive. Often the wear/tear, extra depreciation, etc. is less than 15 cents per mile (unless you have some expensive or exotic car). This includes lost tire tread, chance of a windshield chip, chance of an accident, chance of breaking parts, lower resell value, sooner oil changes, sooner maintenance work, etc.
I drive a lot (about 30,000 miles per year on my car alone and probably another 5000+ per year on my wife's car) and have done the calculations several times on several different types of cars over the years. My cars run closer to 10 cents per mile but I tend to drive cheaper cars to operate.
So $0.15/mile * 850 miles = $127.50. If the rental costs more than $127.50, it likely isn't a good choice. That is, unless you get extra value from the rental (such as the ability to carry more people that must go, etc.) In your example, the rental costs $175 more than driving your own, so it probably isn't worth it. But heck, if you get $50 of enjoyment out of renting the car, do it.
Adjust the $0.15/mile as you see fit for your own vehicle, but at least it gives you a reasonable guess. And remember if an 850 mile trip causes your engine to die, it was probably going to die anyways. You can't just think of the worst case scenario and assume it only applies to this one trip. You need to use average costs for an average trip.