You don't need to represent the actual length of the solution in a finite number of bits, you just need to represent the solution itself. The solution can be represented as the set of edges contained within it. Each edge is either present or not, so a solution can be represented as a bit set the size of the number of edges in the graph, or at most n^2 bits where n is the number of vertices.

For a practical idea of how you'd do this, I think you'd just modify your binary search based on the possible circuit values. Basically, figure out what number excludes half of the remaining solutions, rather than just half of the range, and use that in your binary search.