How many people in all? And how old are the3 small children?
I think 3 nights is enough for Paris, especially with small children. It will give you a taste, and people can return another time if they wish. I see Switzerland as the main focus of this trip, and it is a great destination for multi-generation groups. We took my son’s family, with 6-year-old twin boys, to Switzerland in 2018 and we all loved it. We spent all our Swiss time in small villages (3 nights Bettmeralp and 5 nights in Muerren), and did not stop in Luzern or other cities at all. But that is our style. Maybe you prefer to have more time in a city, . . . Or maybe you should consider 3-4 nights in a village like Muerren (car-free, unlike Grindelwald) and only 2 in Luzern. You can reach the Zurich airport in just over an hour from Luzern, so there is no need to spend a night in Zurich unless you have an early morning flight.
Paris to Beaune would best be done by train, as it sounds like you have more people than will fit in one rental car. And if the children are young, they would need car seats that conform to the French requirements, whatever those are. You will be using trains in Switzerland, so why not start with them right away?
The fastest journey from Beaune to Zermatt takes 5 hours 45 minutes, with changes in Dijon, Basel, Bern, and Visp. There is an alternate route with only 3 changes, but it takes almost an hour longer. It is up to you, but I will say that we have traveled around Switzerland in family groups ranging from 6 to 10 people, taking train journeys that involved 3-5 changes, and have never missed a connection.
I double hotels will be taking reservations for 2022 until sometime this summer, but you could start researching now and build a wish list. But are you sure you want hotels in Switzerland? Apartments or vacation houses (Ferienwohnung in German) are widely available and a better value. Also they offer more room and the ability to make some of your own meals.
When we first started vacationing in Switzerland (20 years ago) the apartments were generally rented out on a weekly basis only, Saturday to Saturday, but that has changed. For our recent trip I found lovely apartments for our 3-night and 5-night stays. And even in 2007 I found a 2-bedroom apartment in a fantastic Zermatt location for our 3-night stay, because we were there before the start of high season (we arrived July and the high season rules and prices began July 5).
As for August versus September, there are trades-offs. You might find shoulder-season pricing and policies in place in September, and perhaps a fall festival to see. But I don’t know for sure, as we have not been in Switzerland in September. We visit the Dolomites in September, and hotel prices there generally drop at the end of August when the kids go back to school.
August will likely be more crowded, and possibly more expensive, but there might be more festivals on offer. We have enjoyed the little Dorffest (village festival) in Muerren on 3 occasions now, and I actually plan our trips around that (first weekend in August). The cow parade is a highlight of Rick’s older TV show on Switzerland (I don’t know if it still appears in his new Swiss Alps show, filmed in 2019).