Wouldn't be wise to work together, seems the disturbance causes their flight to seek cover, knowing its a problem ahead of time, both parties could work together on it for resolution. I don't see how the blame game works so well, nothing gets done and the problem persists.
You want to talk about being invaded by rodents, in '01-'02 I was working on a new building on the NYU campus, adjacent to it, across one street, the site work just commenced for the next building and when they started demolition, the rats started coming out and literally took over the new building. I have never seen anything like it, and the CM/GC had a serious problem on their hand, they never prepared for it. Rodent control starts at the foundation stage and lasts throughout the life of the project, or until its deemed no longer necessary.
These rats chewed holes through new drywall like it was nothing, they made passages, and paths everywhere, the sheer number of them was just incredible. Darned things would ride the elevator with you, smart as a whip and right until the building was to open they were a real problem, was never so glad to get our last gangbox off that job.
I was heavily involved on the next one adjacent to this one, once the steel started going up, and as that one progressed, I never saw one rat in that building, sure they were around, they always are, but nothing like what happened after that mass exodus.
I've had trouble with them on other jobs, but knew that an exterminator is part of the general conditions cost, best to get them in on time and regularly, or have your laborers do the same work needed to control them. They love unkept job sites, so its paramount to keep the entire site clean and orderly. The developer whom I worked for said you run a clean jobsite, I agreed, and said you can't let housekeeping go on any site, just pays to keep after it for numerous reasons, mostly safety, but also pest control. Blaming this and that does nothing.
Barney Fyfe said it best, "nip it in the bud", don't do that and you will get over run, the city is teeming with rats in places.