I'm wondering if it's possible to have LODs and variants as separate sets.
Is there a way to keep my LODs selected at the correct level when I switch my variant?
And so will they keep working if I change the variant?
I've faced this problem many times, but I can't figure it out.

If they are changing different things in the scene, then they shouldn't affect one another. In context of the Component Builder, you can use the Component Geometry Variants node to make your LOD variants (i.e. geometry). And then the Component Material nodes will add to the material variant. And those are separate. If you use Component Material nodes as inputs to the Component Geometry Variants node, then the available material variant selections could be different, when you change the geometry/LOD variant.

It's worth mentioning that USD doesn't have a special notion of "LOD"; most people use Variant Sets to capture LOD options (with the understanding that these can't be animated to change dynamically over the timeline of a shot). If you want to separate LOD variants from other variants, just ensure the other variants are either not affected by the differences between the LOD selections, or you encode the other variant sets within each LOD variant selection.

I couldn't quite understand that file, but hopefully this explanation helps!