September will be pleasant but I suggest you carry enough coats and scarves, just in case. Layering will be essential as the temperature can drop from time to time. You'll can travel between the countries using the rail or flying is always the easiest option.If you'll give five days each to each country, you'll can pick and choose experiences at a slow place.
In France, a visit to Paris would be lovely. You can walk about the streets and gardens, check out the museums like Louvre, Musée d'Orsay, Rodin Museum etc. Visit the Eiffel Tower, eat macarons and drink coffee at Verlet. South of France is always lovely like Nice and Marseille but maybe you can go a little off the usual tourist track. Why don't you head to Lyon and take a wine tour? Or Dijon for more wine and some delicious mustard? The cobblestoned-streets, palaces, gardens and cathedrals will make it an echanting getaway.
Before you head to Germany, go through Luxembourg (spend half a day maybe?) before reaching Frankfurt. The old town and museum quarters are a wonderful way to spend time in the city. You must head over to Rhine Valley where you can take cruises and see nature at its best.
Italy has much to offer but you need to decide whether you prefer the north or the south. Maybe you can visit Padua instead of Venice which is always full of people. Then make your way down to Milan, Florence and Rome. Italy is full of art, architecture, food and natural beauty so you can choose to do what you're keen on.