My experience was not the same as your plan but perhaps close enough to be somewhat relevant. Last month I flew out of Italy to Germany; I went Venice to Frankfurt, not Florence to Munich, and I flew Lufthansa, of which Air Dolomiti is a subsidiary, and economy class, not business. But they did weigh our carry-ons. The carry-ons (which were the Rick Steves "Convertible Carry-on") were within the size limits (not overpacked) but failed the weight test, which I believe was one bag of 10kg (figures that business class allowance would be more generous at 2 bags of 8kg each). We took a few minutes to move anything we wanted en route (we were headed back to the US) into our personal items, which were not weighed. Then we checked our carry-ons through to the US. But need to be careful about personal items, because Air Dolomiti, as a regional carrier, probably flies smaller aircraft that may require stricter weight limits (we were on an Airbus A320). Hope this helps, and have a great trip!