Calculating the number of non-intersecting routes in an Euclidean graph
There are exponentially many such routes.

Think of a sequence of $n$ diamonds.  At each diamond, you can go either left or right, independently of what you do at all other diamonds.  This leads to $2^n$ paths, each of which is non-intersecting.  Now the complete graph on those vertices contains all of these paths, plus some more, so this is a lower-bound on the number of non-intersecting paths.

Therefore, you can't hope to find the optimal one in polynomial time by guessing or enumerating non-intersecting routes.

You probably could have guessed or suspected this, from the fact that Euclidean TSP is NP-complete....