For what it’s worth, some remarks (not in any logical order)...
- Switzerland in august will not be exactly a cold place, unless you go high up in the mountains. Luzern will have summer temperatures.
- If staying 3 days in Luzern, you will need the car (so don’t drop it off there). Do some trips in the area or you’ll get very bored...
- Nice (and the surrounding area) is IMHO overrated and overpriced.
- Avignon: good choice as a starting point for several day trips in the south of France.
- Interlaken: I would consider spending 2 nights there just to take a day trip by train up to the Jungfrau. (This is for me still one of the most memorable trips I have ever made.) But you have to be lucky with the weather on the day you have chosen.
You could also do this from Luzern, which would give you more flexibility based on the weather forecast.
- Italian lakes region: good choice if you want to combine mountains and warm temperatures. This is one of my favourite regions in Europe, although I prefer the “Italian” part of Switzerland (Lugano etc.), not Italy itself.