We've also been a lot, as well as all the above advice, we liked this place May be trickier to get to if you don't have a car.
Roskilde is a good day trip from Copenhagen. Legoland is great, but IMO there's not very much else to see around there- may be an idea to base yourself elsewhere and travel for that 1 day? Aarhus is great to visit, and there's the option of the fast ferry from Aarhus?ebeltoft to Zeeland for Copenhagen.
Marielyst does have a reasonable beach, although I didn't particularly like anything else about it there (but this was pre kids). You may find beaches on Jutland, which is where Legoland is more surf-like.
Last time we went we got a Copenhagen card which we used a lot. We got a 3 day card but staggered it over 4 days, so started at noon day 1. This meant we were able to use it on the day 4 morning, in fact we went to Sagnlandet and had the day there. It's also valid for transport and I think children under 10 are free so you would only need 1/2 adult cards.
As has been said, food is very comparable, even booze. And they have Netto/Lidl which makes it even better.
Depending on your self catering comfort needs, many Danish campsites have little huts they rent out. Some will be basic, think camping pod but many others have bathroom/kitchen etc. May mean you are able to base yourself in more places?
Example here
Danish campsites main site