Solve Travelling Salesman once you know the distance of the shortest possible route
You try to find a tour of length ≤ K. What would likely happen: If K is very large then a tour is easy to find. If K is very small then it is easy to prove that a tour doesn't exist. The closer you move K to the critical point (where a tour of length K exists, but K-1 doesn't exist), the harder it will be to either find a tour or prove it doesn't exist. You don't beat NP-completeness that way. 

And actually, since NP-completeness is about decision problems (problems with a yes/no answer), that's what the travelling salesman problem actually is: Deciding whether or not there is a tour of length K or ≤ K.