The prisoners have gone away, I've collected all but one badge on the next
level, and I learn that the missing badge is on the other side of this
rope bridge. Please don't tell me I've got the big part of the
last level to do again...

There should be an engineering manual on building/repairing rope bridges somewhere in the room the prisoners were in. I believe I found it on the floor on the west side of that small area near some tables.