usually excluding objects from SSR is not possible, because it’s a screen-space effect and there is no information “behind” the object that you want to skip
however excluding the background can actually be achieved. the SSR shader code performs traces in screen space using the depth buffer to then sample the result of the previous frame.
manually modifying the SSR hlsl code, you can limit the screen-space traces to a maximum depth which will be interpreted as a failed trace. and like other failed traces (i.e. the screen borders) the shader will fall back to displaying the skylight’s reflection cube.
so in short, yes you can do it. but you need to modify the SSR hlsl code