The short answer is that there isn't an insurance company alive willing to take on that liability.
The longer answer is that what you're describing is essentially a
fixed-price contract. The actual construction on most large infrastructure projects is actually done under fixed-price contracts. Most of the cost inflation you see comes before that, during the permitting/environmental phase. No contractor is going to take on liability for a project at that stage, cost estimates will still be all over the place.
Contrary to popular belief, private companies don't have better lawyers or some magical organizational efficiency. They just make the local munis take on the cost liability during the riskier stages of development. And the local munis have to accept it, since otherwise no one will take on the risk at all and nothing will get built.