Well, the setup for equirectangular images is a way to work with the distortion when projecting onto your physical sphere. It is a way to get your content to conform to the circular/spherical shape. To map projection to a complete 360 physical sphere requires 4 projectors. When you use 1 projector you can't avoid the distortion of the images as the projection image tappers around the physical shape. The equirectangular image format is likely to assist with dealing with the outer edge distortions. If the distortions are not critical because of the type/style of content you are projecting a simple circular mask might be all you need.
This is an equirectangular image that is distorted at the top and bottom so that it will wrap around a 3D model sphere at the polar regions.
Here is the same image used with a 3D model sphere in Isadora projected roughly onto a physical sphere using one video projector (earth globe is all I had at hand). Because the physical surface is curved there is inherent deformation of the projection surface. The viewing angle of the audience is also going to be optimal at the same approximate position as the projector.
You can pre-render content and then format the render/re-render as 360 video using a number of different software: Blender, PremierePro, Final Cut Pro
But... If your content is more organic or you don't mind the distortions, as mentioned you could use a simple circular mask shape as an alpha mask on your regular rectangular output.