I, too, like to plan my trips with an eye to the historical weather patterns. However, your revised target list for this year covers more ground than I would want to tackle in 15 or even 17 days. London and Paris are monumental cities with nearly endless sights. I would prefer to spend more time in them (certainly not less than 4 days, meaning 5 nights, and really that's not long enough). That would mean further reducing the geographical extent of the trip.
Prague and Amsterdam would both be 3-day (4-night) minimums for me. I think Prague's location screams "flight" (or "two flights"), and I really do not like having to deal with airports in the middle of a trip, especially a trip this short. Furthermore, the city can be quite hot in the summer, so I don't think it necessarily has to be tackled on this trip.
I like Bern, but most people go to Switzerland for the mountains, not for the cities. The usual recommendation is the area around Lauterbrunnen, and it's best to schedule enough time there that you have a reasonable chance of a good day or two for outdoor activities; Swiss weather is extremely unpredictable. So this, for me, would be another 4-day/5-night stop. Swiss rail fares (along with everything else in that country) are quite expensive, so traveling into the country and back out again for a very short visit will not produce a great deal of sightseeing time per $100 spent.
Bottom line: The trip will still be much more rushed than I would like, but dropping Prague and Bern and substituting the Italian Dolomites would probably save some money and allow a bit more time for the big cities. However, travel time between Bolzano and Paris/Amsterdam is considerable: Bolzano is over 10 hours from Paris by train, and you still need a bus to take you from Bolzano into the Dolomites. Flying from Paris to Munich, Milan or Venice might save a couple of hours (or not), but now we're back to dealing with airports. Much as I like the Dolomites and think they are a great summer destination, there's no way I'd be willing to give up two days of a two-week vacation to travel there and back. That area works much, much better as part of a trip that includes other destinations in northern Italy.
I think three of London/Amsterdam/Paris/Normandy (or even two of those) would make for a better two-week trip. There's be a lot less time wasted on traveling between stops, so you'd be able to see a lot more. There's lots to do in Normandy, and there are attractive towns easily visited from Amsterdam, London and Paris.