William has referred to his wife as both Catherine and Kate. I'm thinking in particular of the engagement interview, where he refers to her as "Kate."
The media will continue to call her Kate Middleton for a simple reason; name recognition. Name recognition leads to clicks on an article, and hence more money.
Case in point is my 12 year old son. A local radio station mentioned Kate Middleton the other day and the royal baby, and he interrupted, asking "who is Kate married to again?" I was surprised that he even vaguely knew who she was, but he does know and has enough interest in her to realize that she and her baby are important, hence he inquired about the story. He's really a fairly good example of John Q. Public. Had he seen an article about Kate Middleton, he would probably click on it, scan the first paragraph or two, look at the photos and move on. I'm fairly sure that if he had seen the same article on "Catherine, Duchess of Cambridge" even with her photo, he'd pass it by without a second thought. Point is, we royal watchers will always click on an article on the royals, no matter what the headline says. The general public won't.