I just had a guest recently who saw mouse droppings and not only did not complain, but set a snap trap for me and disposed of the mouse and STILL gave me a glowing review. BUT my place is in the Catskill Mountains of New York State. As it happens - the following weekend we found out where the mice were getting in - they chewed through my dryer hose - and remedied it.
This very issue has been a worry for me, but fortunately it hasn't happened as the OP described ... not yet anyway. I have seen other hosts in the Catskills specifically outline that 'woodland creatures will occasionally make their way into the house and if this bothers you, this is not the property for you' and I may add language to this effect, but I bet Airbnb would not honor it. Last winter was extremely mild in the Northeast, which means the rodent population just exploded. As things warm up all over the Northeast this will be more and more of a problem. Besides, no exterminator in the Catskills is going to guarantee you won't get more mice. No way. The locals laugh at city people who hire exterminators to get rid of mice.
PLUS, if I were to rent my New York City apartment - which I can't and wouldn't because I rent and its against the law - that could be an issue here, as well, because it is an old, old building and frankly, mice are an on-going issue in any old NYC apartment.