Environmental light shines through the skybox. That's the point.
Where would sunlight come from without a skybox?
So why do you need an environment light? That's meant for light coming from the skybox.
Make the skybox texture not sky, but use the skybox material. Proceed to use light_environment.
Use a skybox of some sort. It doesn't need to look like a sky. Have light come through that.
You can make new skybox textures, no? But light comes out of the skybox material, not the texture. So you get your ambient lighting like you asked for, but there's no visible sky.
Again, I really don't understand what you want to do. Environment light comes from the skybox. That is the only purpose of that entity. If you don't want light to come from the skybox, use a different entity.
So put a light in the room. Scale it so that it doesn't appear to be coming from a single point. If you have difficulty doing this from the size of the room or something, add a lot of dim lights everywhere with very large maximum distances.