Run it by your kids to see if it interests them, but I would suggest a visit to Casa dels Entremesos, Placa de les Beatles 2. it is within a couple hundred meters of the Cathedral, is free, and would take about 45 minutes to an hour to visit. Check it out at www.lacasadelsentremesos.cat or google it. Many of the “giants”, beasts and big heads that are traditionally paraded through the city during festivals are housed here. There is also a short video of some of the festivals featuring the giants, many of which are 10-12 feet tall and weigh up to 150lbs. Interesting history associated with them.
There is a good website called barcelonalowdown which has lots of great suggestions for things to see in the city. Are you only in Barcelona for one weekend? If so you really won’t have time for much besides your “definitely doing” sites”, and they all require advanced reservations.
My other suggestion would be to take the kids to the Barcelona History Museum, where they could take an elevator down to view and walk through the ruins of the underground Roman city Barcelona is built on.
Personally I would present the kids with options (or have them do the research, nothing like vested children) and let each one select something that they are interested in. For example they might like the Maritime Museum if they are interested in old medieval boats. Sports? The Olympic Stadium or Camp Nou. Dance? Sardana dances in front of the Cathedral on Saturday. Beaches? You’ll have your pick.
I would only take kids to the Picasso museum if they really, really want to see it, as it is very crowded usually, and doesn’t strike me as a place that would be of interest to many young people. Same with Miró. The MNAC is great, but huge. Unless your children are art lovers I would probably not drag them through any of those places, wonderful as they are.
If you do just have two days I would go to your listed must sees and spend the rest of your time wondering the medieval streets of the older parts of the city, relaxing in the squares, and soaking in the atmosphere. Enjoy!