What's the meaning of Farmover type, as in the following paragraph? "Richard Caramel tries to move about as if his figure were better. He is torn between his innate cordiality and the fact that he considers these girls rather common—not at all the Farmover type."

The word "Farmover" appears to be a combination of Farmington and Westover. 
Farmington is the location of the prestigious Miss Porter's School, an all-girls college preparatory institution. According to its website, the school was founded by Sarah Porter, whose focus on female students' academic excellence and participation in sports was revolutionary for her time. Notable alumni include the cream of American society such as Jacqueline Bouvier Kennedy Onassis (former First Lady of the United States) and Letitia Baldrige Hollensteiner (former White House aide and social secretary to Jacqueline Kennedy). Other alumni are movie producers, actresses, CEOs of multinational companies, and professors at institutions such as Harvard.
Meanwhile, Westover refers to the elite Westover School, located in Middlebury, Connecticut. Like Miss Porter's School in Farmington, Westover School is an all-girls college preparatory institution. In the book, the word "Farmover" is mentioned three times. The quote you mention above is quite revealing.
In the quote, Richard Caramel does not think very much of Miss Kane and Miss Jerryl, women he considers "common." In his mind, they are simply not "Farmover" women, the products of elite, exclusive boarding schools. Many graduates of these schools also move in the most prestigious circles of society. What Richard Caramel is saying is that Miss Kane and Miss Jerryl are beneath his notice.
This is, of course, very biased judgment on Richard Caramel's part. However, Caramel's conclusions may have been influenced by his creator, F. Scott Fitzgerald. Fitzgerald's first love, Ginevra King, attended Westover School herself, and Fitzgerald was so smitten with her that he modeled The Great Gatsby's Daisy Buchanan after her.
Additionally, it might interest you to know that Fitzgerald spent the summer of 1920 in a town called Westport in Connecticut, where he finished writing The Beautiful and Damned, the novel your quote originated from. Connecticut is significant because it is rumored that Fitzgerald modeled the Gatsby mansion after a particular home in Westport. For more, please refer to the links below. Suffice to say, the use of "Farmover," a reference to two boarding schools in Connecticut, is no accident on the author's part.
http://www.princeton.edu/paw/archive_new/PAW03-04/04-1105/feature1.html

https://patch.com/connecticut/westport/f-scott-fitzgerald-slept-here