The movie Elvis & Nixon has quite a nice score, but does not feature any music by Elvis. Why is that?
The licensing costs might be an issue, but it's more likely to be because of the required approval from CKX Inc., "the company that controls Presley's music, image and other intellectual property"
The following quotes are from this Hollywood Reporter article from 2011 (emphasises are mine):
(...) the famously tightfisted CKX, which acquired 85 percent of Elvis Presley Enterprises from Presley's daughter, Lisa Marie Presley, for $100 million in 2005, must approve all uses.
(...) CKX's fierce protectiveness of the Elvis image and the cost of licensing music, which can run into the hundreds of thousands of dollars per track (...)
Then, for Elvis & Nixon (which I haven't seen) add the fact that it portrays a "darker" part of Elvis' life and fictionalises some parts:
The company (CKX) has a financial incentive to maintain the image of an almost cartoonishly feel-good Presley whom many fans remember, meaning it likely would not endorse a warts-and-all biopic.