Considering a run of a Variant is similar in lenght to a "Normal" dungeon, I really doubt this can be done.
Variants sound like 3x the work than a regular dungeon. What you ask is impossible. Would it be cool? Yeah. I wish we had at least a daily roulette for Variant dungeons (or for BLU dungeons, but that's another thing entirely...).

In short, cool idea, will never ever happen.