Not tried the other two but would definitely recommend Rome or two loved all the sites and DS enjoyed helping navigate around all the attractions. We saw all the sights and had a lovely morning in the gardens on an electric bike for the 4 of us. Barcelona is very nice too but avoid high summer for both places.
Having been to all three (although not with kids), baring any particular preferences on your families part, I would say Lisbon. It's a lovely and fascinating city, and the food is better than the others IMHO.
I can't comment on Lisbon and Copenhague but I lived very close to Rome for 7 years when DS was 4 -11.
It can be very hot in June, not unbearably so by Rome standards , but still hot (30-35). If you are looking for things for the DCs such as science museums, art museums that are suitable for young children, interesting playgrounds, then Rome isn't what you want. However Rome can be a great city for children, my DS loved Rome.
I think if you go to Rome with children of that age you really have to prepare them for it and plan well, especially if you only have 3 nights, because the wonderful sights they can see are a bit meaningless without it. You have to explain to them, or show them a couple of films about Roman history and the Roman Empire. If they know what they are seeing it is a fantastic city for kids. When they are there you can say you are walking where the emperors walked. You can go into the Colosseum and say this is where the gladiators fought. After showing them Ben Hur you can visit where the real chariot races were. You can explain to them the legend of the Mouth of Truth and dare them to put there hand in. That only slightly touches the surface of everything there is.
Rome is an absolutely wonderful and beautiful city but it takes a bit of effort for young children to enjoy and appreciate it. If all else fails though, they will love the ice cream and pizzas 😊
Copenhagen would be a good temp and has Tivoli, big aquarium, nice walks, good shopping, nice food, everyone speaks English.
I haven't been to Lisbon and I've heard it's lovely, so this would probably be my choice. Yes, it's far south, but Portugal usually gets nice breezes off the Atlantic (and it wasn't too hot in Porto in June).