Some decent snap traps, under boxes with 2-3 rather small holes if not in a complete closed shed so that no other animals (birds!) can get to it. Baited with hazelnut waffels, peanut butter, whatever. A wildlife camera helps to see if there are any, and where, what they do etc.. I know many don't like the killing but imho if they are around the rabbits it's not the time to be squeamish.
The council most likely will put lots of poison boxes everywhere, so the rats are going to die anyway, in a not very humane way, and mice, and whatever eats the poisoned rodents. I prefer to kill any that get too close to my rabbits quickly, and I have a deep aversion against poisons.
Talking with the neighbour sure is a good idea, would be good to solve that issue at the root.