Length of the solution calculation in Breadth-first
I do not know how your sources define "route cost" (I've never heard that term).

There are two points of view I can imagine.


Length of the solution

Once you reach a fitting leaf you have found your solution and can
   abort. The length of the solution, i.e. the number of steps you have
   to perform to get everybody across, is the length of the path from the
   root to that leaf.
Cost of the algorithm

The number of steps the algorithm has to perform is different, of 
  course; it may visit many nodes that are not part of the final 
  solution. The details depend on the specific implementation at hand.

Specifically, you seem to be concerned about backtracking out of
  dead ends. The usual way to implement BFS is using a queue which
  makes backtracking free since you don't actually do it; the next
  node to visit is already at the front of the queue.

If you don't use a queue and backtrack explicitly, well, you get 
  quadratic worst-case performance. Don't do it.