After talking to Duncan you learn he thinks there are thirsty cats inside the Establishment and he wants to sell them milk.
Duncan will leave if you remove the sign. You can access the sign from the rooftop.
There are no switches or buttons to control the sign from here.
Maybe you don't need to turn the sign off. It could be also covered with something.
You will find an item large enough to cover the sign from some other place. Keep investigating.
You will find a blanket.
When you enter Le Boule de Poils you will find a blanket. ?
Take the red blanket covering Oscar the cucumber on the stage.
So, as the sign is now burning talk to Duncan. He will leave.