Probably the simplest way to search for an image with the highest probability of being a cat is to use a technique similar to Deep Dream:
Load the network for training, but freeze all the network weights
Create a random input image, and connect it to the network as a "variable" i.e. data that can be changed through training
Set a loss function based on maximising the pre-sigmoid value in the last layer (this is easier to handle than working with 0.999 etc probability)
Train using backpropagation, but instead of using gradients to change the weights, back propagate all the way to the input layer and use gradients to change the input image.
Typically you will also want to normalise the input image between iterations.
There is a good chance that the ideal input you find which triggers "maximum catness" will be a very noisy jumbled mess of cat-related features. You may be able to encourage something more visually apppealing, or at least less noisy, by adding a little movement - e.g. minor blurring, or a slight zoom (then crop) between each iteration. At that point, it becomes a more an artistic endeavour than mathematical.
Here is something I produced using some TensorFlow Deep Dream code plus zooming and blurring to encourage larger scale features to dominate:
Technically the above maximises a single internal feature map of a CNN, not a class probability, but it is the same thing conceptually.