Notice that as the flap is deployed, both the camber and the area increase. This change in wing shape modifies the lift curve as follows:
(source: Development of a Fowler Flap System for High Performance General Aviation - Wentz (NASA))
Note that the lift curve slope has increased.
Here is my analysis: If Fowler Flap is deployed, the planform area increases. If the planform area increases, the aspect ratio decreases (since wing span is unchanged). If AR decreases, then the lift curve slope should decrease (see How does the aspect ratio of a wing impact its lift? for details).
So where in my analysis do I go wrong? I know the lift curve slope should increase, then why does my analysis indicate otherwise?