Finding if a graph is still strongly connected after edge removal

As Sneftel points out, the distance labels can only increase. If u no longer has a path to v, then I guess v's label will be infinite, so the sum of labels will change from finite to infinite. Yet the sum can increase without the graph losing strong connectivity, e.g.,

u<----->v
 \     /|
  \|  /
    w

where v's label increases from 1 to 2 because of the indirect path through w.