LightMix doesn't affect the other render elements, which will always act to composite the Beauty rather than the LightMix. In general it's taken that if using LightMix, you won't then also be compositing and adjusting the composite in post, but using the final result from the LightMix element.
If you wanted the other render elements to work, I would envisage this kind of workflow - create scene, set up the various lightmixes and save them; save the lightmix passes as images just because; now go through each saved lightmix, use >scene, and save a new version of the scene, with that lightmix now baked in place, and render, to get the other render elements based on those light intensities (including HDRIs, with certain ones now disabled); repeat for each lightmix you created, creating a different version of the scene and saving it and rendering to get the render elements for that particular look.
Of course you could skip saving the scene multiple times, just load original scene, load lightmix, bake, render, save render elements, if you wanted to save the space of having multiple versions of the scene saved.
I think that would do what you are looking for, if I understand correctly!