I did Rick's Best of Italy starting Sept 30, ending Oct 16. Weather was variable, some rainy days but not too many. I will tell you that I was more affected by the heat in Rome at the end of that time. Your profile doesn't reveal anything about where you live so if you are intolerant to heat, I'd go with your original dates. I think that this Fall was hotter than "normal" whatever that is any more, lol. For myself, I'd be more concerned about heat than rain.
IF you do go that late in October, I'd start in Milan and do the Dolomites early on as they will be the most impacted by weather, I think. When we spent our 2 nights in Castelrotto (Oct 2-3) it was very pleasant temps but some of the seasonal restaurants and activities were starting to wind down.
I'd also agree with the recommendations to trim your wish list. The Best of Italy covers everything on your list in 16 nights except Pompeii/Naples and that was moving very quickly with a guide and bus driver who are familiar with the areas and the hotels and can get you places quickly. Doing it on your own will mean more time spent in transit, locating your hotel, figuring out public transit if any, etc.