6-8 weeks for a family of 4

Your post states both 6-8 weeks and 6 months for your timeframe. Which is correct as that will factor into the suggestions you receive?

Are you driving or using public transportation (i.e. trains)? Do you want/need a loop itinerary, either because of airfare or rental car returns? Or can you do "open jaw," i.e. in one airport, out from another? Do you have an airport(s) in mind, given that you are focused on costs? And yes, please clarify if it is weeks or months. If it is months do you need to consider the Schengen rules for American travelers? Oh, and what months specifically? You mention the overlay with summer camp, but that could be anywhere from May to September depending on where you live...

Where have you been in Europe so far?

How much time?

Are you clear on the Schengen limits of 90 days out of 180? When you say "6 months", this is far longer than Schengen allows. You either need to limit the trip to 90 days, or have a visa which exempts you from Schengen limits.

Before any planning, you need to get clarity on this issue.

They say 6-8 weeks two times; I am thinking that “6” before the months is a type, and they really mean 2 months. But they should confirm.

Yes, the "6 month" was a typo. It's 6-8 weeks, in June, so Schengen isn't an issue. Whether we drive or travel by train will depend on what is cheaper as a family of 4. I don't need it to loop.

one of the possible itineraries was Dubrovnik, Ljubljana, Venice, Interlaken, Chamonix.

So far I've been to:
Budapest, Halstatt, Salzburg, Munich, Cesky Krumlov, Prague, Vienna, London, Bath, Oxford, Paris, Amsterdam, Geneva, Chamonix, Pompeii, Naples, Capri, Rome, Florence, Grindelwald, Gimmelwald, Bilbao, San Sebastián, Madrid, Tenerife, Brindisi, and Corfu.

Updated to add: Copenhagen, Malmo, Gothenburg, Stockholm, Bergen, Oslo

And just since we’re at it for other xUS places, Tokyo, Kyoto, Osaka, Hiroshima, Naoshima, Teshima, Koyosan, Vancouver, Whistler, Banff, Glacier Ntl Park, Jasper Ntl Park, Revelstoke, and San Ignacio

Good list.

I have seen Dubrovnik mentioned in some “don’t go” travel articles, due to “overtourism”—-too crowded , especially with partners, and their disrespectful behavior. Just so you know.

Venice is on those overtourism lists, but I would still go, and treat her gently and with respect. We spent a month there when we first retired, and have been back several times since, for a week, or only 2 days. I am still captivated by her charms, even with the crowds. We have learned how to avoid those.

Ljubljana is a wonderful little city with a nice oldtown, a riverfront promenade, an old castle perched above, and interesting history and architecture. English is widely spoken and visitors are given a warm welcome.

I will even suggest an apartment, down a little alley right off the riverfront promenade, in the oldtown. We were very happy there. They offer several apartments in the same building.

https://mescanka.si/

Why Interlaken? It is not the most charming place in the Berner Oberland. You have been to Grindelwald and Gimmelwald, so you know what the area is like. Maybe Mürren. We have spent nearly a week there several times with our grandchildren that age. They loved it, especially the playground at the top of the Allmendhubel lift, where they have spent hours, once when they were 6, then again this past September at age 11.

https://www.myswitzerland.com/en-us/experiences/flower-park-the-fantasy-alpine-meadow-at-the-alp-allmendhubel/

Or you could stay in Wengen and ride the lift up to Männlichen, where they will find this playground, with a slack line, obstacle course, slide down a cow’s tongue, and other fun playground attractions:

https://www.maennlichen.ch/en/summer/experiences/alpine-herdsmans-playground.html

Both Mürren and Wengen have public pools the kids can use (and adults as well). I believe the Wengen one is outdoors. The Mürren pool is inside the Sports Center. Some apartments will proved a card for free or reduced entry into the pool.

We like Chamonix a lot too. Our guide from the Tour du Mont Blanc lives there with her family. She works for the certified guides of Chamonix, a company that has lots of good day by day programs for kids of all ages.

https://www.chamonix-guides.com/en/activities/category/juniors

One route in particular comes to mind:

Nice > Genoa > Turin > Sirmione (Lake Garda) > Venice

Nice has good airport service to begin the trip, along with the beach, parks, a steam train up into the mountain (Train des Pignes), amusement parks nearby, etc.

Genoa is an often overlooked city (by American tourists anyway), but has the biggest aquarium in Europe. Your oldest is probably already starting learn about Christopher Columbus, so there's a good educational opportunity, along with maritime history, a children's museum, and great food.

Turin has the auto museum which is awesome for kids and adults. It also has nice parks (including a fake medieval village) and a good, modern/cage-free zoo just outside the city. Lots of outdoor activities in the surrounding enviorns. Oh, and Turin is known for its chocolate!

Lake Garda also is a great outdoor destination, with activities in and by the water. Good concentration of amusement parks and water parks too, castles, and gardens. Americans probably know Lake Como better, but Lake Garda is a popular European vacation destination and has a very family-friendly infrastructure accordingly.

And finally Venice, which speaks for itself!

So far I've been to:
Budapest, Halstatt, Salzburg, Munich, Cesky Krumlov, Prague, Vienna, London, Bath, Oxford, Paris, Amsterdam, Geneva, Chamonix, Pompeii, Naples, Capri, Rome, Florence, Grindelwald, Gimmelwald, Bilbao, San Sebastián, Madrid, Tenerife, Brindisi, and Corfu.

Here are some options:
Spain and Portugal:
Barcelona, Valencia, Alicante, Granada, Rhonda, Seville, Cordoba, Lisbon, Evora, Cascais, Sintra, Obidos, Coimbra, Porto, Pinhao and the Douro River Valley.

Turkey and Greece:
Istanbul, Athens, Corinth, Delphi, the three islands near Athens, Greek islands of Mykonos, Santorini, Rhodes and Crete.

Scandinavia:
Look for a cruise up the coast of Norway all the way to the North Cape (must do in Summer). Also, consider Stockholm, Sweden and Copenhagen, Denmark.