We were in the MK last 4th July. First Independence Day since 9/11. The park was jam packed but the atmosphere was terrific. Since we are visitors to your country, we're from Scotland, Americans kept asking us why we had decided to travel since 9/11 when it would have been safer for us to stay in Britain. If money permits I would certainly spend some portion of the day in a park.
Hope you have a great time whatever you decide to do.