As I mentioned somewhere else in this thread, the binary search is not a reduction, because you need to use TSP-decide a polynomial number of times (and for a reduction you reduce to one problem instance).
What you show is that the problem of finding the lowest cost is in FP^NP (the class of function problems solvable in polynomial time with the help of an NP oracle). However, by taking your idea a bit further one can show that TSP is in FP^NP. In fact, it is known that TSP is FP^NP-complete, i.e. it is one of the hardest problems in that class (and thus likely not in FNP, the NP for function problems).
What you show is that the problem of finding the lowest cost is in FP^NP (the class of function problems solvable in polynomial time with the help of an NP oracle). However, by taking your idea a bit further one can show that TSP is in FP^NP. In fact, it is known that TSP is FP^NP-complete, i.e. it is one of the hardest problems in that class (and thus likely not in FNP, the NP for function problems).