"Tears and Laughter: The Joan and Melissa Rivers Story" (1994). Although this isn't a film with the real Coral Browne in it, it has her portrayed by actress Sheila Moore. It's described at Yahoo Movies as:

She became autobiographical and serious in a way America had never seen from her. There had, for some time, been rumors that Rivers and her daughter Melissa Rivers (nee Rosenberg), an MTV and cable reality programming host, had been estranged, and the persistent stories that Rivers had not been a weeping widow haunted her. Rivers and her daughter wrote and starred in "Tears and Laughter: The Joan and Melissa Rivers Story", a TV-movie (ironically for NBC) that dramatized the events surrounding Rivers' firing from Fox, Rosenberg's death, and the estranged mother and daughter finding common ground. The 1994 movie was a ratings success and ended the undercurrent about Rivers being an unfeeling witch.

I don't believe it's out on video or DVD.