It looks as though there is one direct TGV from Amboise to Avignon on weekdays.
I don´t find any direct trains from Amboise to Avignon during October and TGVs do not serve Amboise. Basically, Amboise to Avignon requires a return to Paris Austerlitz on an Intercité train, a station change to Gare de Lyon, and then a TGV to Avignon. I would plan 6 to 8 hours depending upon the train selected.
consider that 3 days of touring empty Renaissance palaces is likely to try anyone's enthusiasm.
I might agree if that is all you did. However, the Loire Valley has more for children than practically any other destination on the proposed itinerary. Consider:
Grand Aquarium de Touraine
Parc des Mini-Châteaux
A large number of bike trails
There are also visits to troglodyte villages, mushroom caves, horseback riding and these are just the obvious activities. A little investigation would probably open many more options.
What children typically do not enjoy is the confinement of lengthly car rides and train trips. By the time a family of 5 packs and leaves one hotel, travels to a new location, finds a new hotel and settles in, most of a day will be consumed. Available daylight in October is noticeably less than in June, July, or August. You do not really want to waste valuable vacation time traveling.
This itinerary needs to be seriously cut back to no more than three destinations and Paris is one of three.