You can set a Flight Alert on Expert Flyer for the flights you want - you're looking for UA Saver Award space (presumably Business - I) - and it will notify you the moment a ticket becomes available... THEN...you can jump on to LM and book for 60k per ticket one way + fees.
EDIT: Quick thought - I rarely see UA Business Saver space on UA to Rome. Most of the Saver Space I see is to MXP via EWR. Not sure if that helps you at all, but just a thought if you can accommodate.