I did this tour Sept 30-Oct 16 2022. It was not too cold in the Dolomites, in fact I think I was in short sleeves the day we hiked up in the Alpe di Siusi (jaw dropping!). It WAS too hot for me at the end of the trip in Rome. That Fall was, I think, warmer than "normal" (what is that any more??) and was in the 80's in the city. I don't tolerate heat very well and I'm also pretty cold tolerant. We did have quite a bit of rain but it did not really impact touring for me. Be sure to have a waterproof rain jacket with a hood. No need for rain pants or waterproof shoes in my experience (not like the Scottish Isles!).
https://community.ricksteves.com/travel-forum/trip-reports/best-of-italy-sept-30-october-16-2022
In years past I've been in Italy the 1st week of Sept (Venice, Florence, Rome and CT) and it was hot. I've been later the same time frame in Late Sept/early Oct but was not as far south as Rome so it was not quite as toasty. We ended in Milan in mid-October and hit rain.
I think your choice depends on how well you tolerate heat. I don't think October is too late for any of the tourist areas although I'd probably want to make sure the Dolomites were 1st - 2nd week of October.
BTW, in your planning, I'd suggest a couple of nights in Milan before the trip and I'd also recommend an extra night in Varenna. You are really only there for 1 full day and there is more to do around there than that. I'd also suggest a night or 2 at least in Rome at the end. I only stayed 1 extra night so I had an extra full day in which I did the Domus Aurea (Nero's Golden Palace) and the Borghese Gallery but I'd been to Rome a couple of times before.
This trip is an excellent sampler!