Dude, this would work if people were any good with numbers. Basically, if you have a competition, the best liar who low-balls their estimate gets the job. And then due to the sunk-cost fallacy, manages it to completion.
As it is, a honest estimation will only lead to your project never being funded, no matter how worthwhile it may be. And frankly, seeing the benefits of funding fundamental research, that would be immensely more costly for society in lost opportunity than whatever cost overrun. So the socially responsible thing to do is to lie about the costs.
Alternatively, you could teach people and their representatives to understand numbers and not freak out when they hear ONE TRILLION DOLLARS. 'Cause absolute values mean nothing. Fat chance.