What is a suitable Tensorflow model to classify images into foggy/not foggy?
From the sounds of the problem you could probably do some thing with extracting some features from the images such as how many edges they have, brightness (to get day/night), average color values. Then using a more simpler classification algorithm such as SVM, KNN, Decision Tree, Random Forest.

The Tensorflow ready-to-use models look to be very complicated models with a large number of layers so it will take a long time to train and run. It will also be very hard to train them from scratch (Retraining a pretrainid network could help with that though). So I think they might be a bit overkill. Also note that those models were probably made with the imageNet dataset in mind which has 1000 classes where you have just 2.

Its very hard to know what will work without seeing the images or being able to trying it first.

I would start with simpler faster methods before trying slower more complicated methods.
So in order try the feature extraction plus classifier if that is unable to learn a good relationship then move on to a basic CNN if that doesnt work move on to the more complicated CNN models.

With respect to the objects within rather than the overall appearance. For a CNN using different pooling layers can effect this change this. eg. Max Pooling will get the max value of a filter so can be largely effected by a small part of the image where as Average Pooling will be better at looking at the whole image.