They wouldn't have used studio lighting, but a military anti-aircraft search light, the brightest possible man-made light source at the time. It was likely used in a studio using a ceiling mounted convex mirror to increase the diameter of cone of the light coming from it, therefore more closely mimicking the light saturation of the sun. The light would shine from the light on the ground into an angled convex mirror mounted on the ceiling, which would the reflected light onto the "moon" scene.

Amateurs from the "Lunar Program" against the cameraman L. Konovalov (pdf)