Please advise on best travel route?

As to how to get there, we have always driven. But if you get there, your family will love it! Give it at least 5 full days (6 nights). And you will need a car when there. We took our kids and grandkids there in 2023 (Kids were 9 and 12 at the time) and it was perfect for all of us. With your 10 days in Paris it looks like your 6 weeks will be all accounted for.

Thanks Judy. I thought about driving but that brings up the issue of booster seats. I've read that it's hard to rely on renting them from car rental companies, but I also don't want to lug them all over Europe from home. I would need 3, one for each kid. Did you just buy them in Europe when you rented a car?

We've never had trouble with renting booster seats in Europe (or elsewhere). There is always a caveat in the rental agreement that it isn't guaranteed, but they've always been available. I usually had some idea of where the nearest place to go buy one quickly is, just in case. You know your own children best, but will the eleven-year-old really need one?

Train Strasbourg to Interlaken West, one change in Basel.

Reaching Dordogne is too much of a headache for me. You will spend all day getting there (around Sarlat) from the Interlaken area, with many train changes. Renting a large car in Switzerland and dropping off in France is very expensive option. Flying does not seem to be a good alternative either.

In theory you can take a train to Geneva airport, one change in Bern. Take a taxi from the Swiss side of the airport to the French side and rent a car there. Timewise, still all day, unless you stop off somewhere for the night.

Autoeurope allows to you to input any number of booster seats on their order page, though you'll have to get in touch with them to confirm.

https://www.autoeurope.com/

Thanks HK. It's my understanding that all children need some booster/car seat until age 12 in Europe. At least that's what I read. He doesn't need one at home, but I wanted to be sure we're following the rules where we are

That was my worry gunderson. I've read such amazing things about the area and how great it is for kids, but I feel we might be wiped with that much travel by the last quarter of our time there. I don't know when we'll be back in Europe so it's hard not to want to squeeze everything possible in this trip.

Family of 5, I shouldn't have said anything. Now that I think about it, we haven't driven with a child in that age range in France. In some places, height and weight are used in conjunction with age to determine when a child needs a booster seat. We've definitely put a child in a booster seat abroad who had aged out or outgrown the seat in the US.

In 2023 with our grandkids (9 and 12 then) wit didn't occur to us to get booster seats, and the car rental place didn't say anything. I honestly think it has to do with the size of the child.

According to the Man in Seat 61, there is a direct TGV between Strasbourg (or Lille) and Bordeaux, which is a good place to rent a car for Dordogne.
Re: Brussels. Really? I recommend heading out to the coast to Ostend or DeHaan. Easy day trip to Bruges (not sure about Ghent) and LOTS of interest to your age range, including an aquarium, the Atlantikwall Museum, the beach. Best of all, no car needed because there is a tram that runs the length of the coast. This would be a nice break from loud, dirty London.
If you're concerned about being burned out, maybe go to Dordogne first, by taking the Eurostar to Lille and transferring to Bordeaux (that's a long day) or take the train from Lille to Bordeaux after Belgium. It's a little zigzaggy, but it's not impossible.
With kids this age, I would prioritize Dordogne over Brussels and Strasbourg. There's just so much to do there that appeals to kids. And it's stunningly beautiful. Don't stay in Sarlat. Just go for the day, not necessarily on market day, either. There are markets all over that area that are not tourist traps.
If you do go to Dordogne during or after the first week in July, be sure to drive through Abjat-sur-Bandiat, as this tiny village will host the Felibree, which is a festival (for what, I'm not sure) that involves decorating the whole town in flower garlands (I think there's a mix of fresh and artificial flowers). I happened upon the remains of the flowers when I drove through Saint Cyprien in September (two months after the festival!) 2018. It was really cool.