In two Los Angeles scenes in a glass elevator, a logo for consulting company KPMG is clearly visible on a building in the background, in the center of the frame. KPMG was created in 1987, when KMG merged with Peat Marwick. The building, now called KPMG Tower, was constructed in 1983.
While Bobby Riggs is watching television in the Rolls-Royce, he passes channels showing Mary Tyler Moore and Kojak. Both shows were on CBS, but on different days, and neither was in syndication in 1973.
While it is true that Billie Jean King was having an intimate relationship with a woman named Marilyn Barnett during the time of the "Battle of the Sexes", Barnett was not a hairdresser King met by chance. Barnett actually worked for Billie Jean (Barnett was King's secretary.)
The images on the monitors in the TV production truck just before and during the final match are clearly simulated. Images shown on professional broadcast monitors of that era either had rounded corners with no gray areas around image, or rectangular corners surrounded by gray areas, but never rounded corners surrounded by gray.