Access itinerary here: https://www.rome2rio.com/trip/vulyhsre
One option that may be as affordable as driving and stopping in Berlin or another German spot is to start in Copenhagen, drive to Billund, then fly to Prague. We could also stay halfway between Copenhagen and Billund because the hotels in Copenhagen are quite expensive. Then we'd take a flight from Copenhagen to Prague. Two attractions within Copenhagen are Tivoli Gardens and Amalienborg Palace, where the guard changes each day at 12 p.m., as shown by the soldiers below.
Copenhagen hotel price: 1 night: $129, Danhostel Copenhagen, private bath, two twin beds, two bunk beds.
We arrive in Copenhagen at 15:20 on June 7. We can spend one night here, or two until June 9, before departing for Billund.
Billund is home to Legoland. Unless we are able to schedule a private factory tour, we will spend one day here at the amusement park.
ETA: June 8
Accommodations: Billund hotels and farmsteads average about $175 per night. One night would be preferable.
Getting to Prague: Rome2rio.com estimates travel to Prague from Billund to cost $82-135 per person. If we fly from Copenhagen to Prague, the estimate is $41-121 per person. Deduct the price of staying in Berlin or another half-way point and a rental car and it could be about the same.
ETA: June 9 or 10
Accommodations in Prague are more than $200 per night. Recommend staying 2 nights minimum.
An additional stop or a side trip is Cesky Krumlov, Czech Republic. It's the smaller, quainter Prague I read about. It is about a 2 and one-half hour drive from Prague and can be the point we go to between Prague and Vienna.
Prague to Vienna by car takes 3 hours, 36 minutes. Vienna is looking less gorgeous when you look at Prague and Budapest pictures, but it's nice.
ETA June 13-15, stay one night or two
As far as activities, I think the kids would love the Turkish baths in Budapest. And we'd all enjoy a cruise on the Danube River.
Accommodations: Agape Aparthotel in Budapest is $133 for two nights! That's half the price of Vienna or Prague. It's also possible to stay in Budapest and day trip to Vienna. Or vice versa.
Flight from Budapest to Larnaca is $121-175 per person.