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).