Hi Kevin,
I had the same problem when i was trying to book a room for mid week dates. The problem is Livigno is a big package holiday destination which run from sat-sat and many of the places i contacted said they need to keep their rooms free for the full week to accomodate these customers. If you look up ski holiday web sites, look at what holels are still available for the full week and contact them directly you might just get lucky and they will give you a room for a few days. Good luck and enjoy your self!