The driver is Stirling Moss. The car is painted and carries number37 so it is the second time he drove the car, ie at Silverstone on 7th May 1955. In My cars, my career by Stirling Moss and Doug Nye it is described as a Beart-Climax.
However, the text says that Francis Beart had the car built for him by Bernie Rodger. In the Robert Edwards biography of Stirling Moss it is listed as a Beart-Rodger - Climax . Martyn Krejnik lists the car as a Beart-Rodger.
So I expect the name you are looking for is Beart-Rodger-Climax.