What you read about summer being one the busiest times to visit is outdated. It is not that summer has low crowds. But crowds are not nearly as bad as times like Spring Break or Holidays.
In any case here are some guidelines for you:
1. Go after July 4. The week before will be very busy because of World of Color and unblocked APs - see
Understanding AP Blockouts and Impact on Crowds
2. If you only have one day you want to choose a day that is not a Magic Morning (MM). MM is on Sunday, Tuesday, Thursday and
Saturday. MMs allow some visitors into DL early and if you come for the day you will be behind several thousand people already in the park - see
Magic Morning (Early Entry) Super Thread. Note that if you arrive late to the parks (late as in after 9AM in July) then it will not matter whether you go on a MM or not. It really only matters if you go near park opening at 8AM.
3. It is better to go on a weekday in summer than weekend - usually.
4. July 4 is a holiday - it sounds like you know that. Since it falls on a Sunday this year then most business will close on July 5 freeing up people to visit around then. Historically the days around July 4 are very crowded at DLR. But last year crowds were low on and around July 4.
I think Wednesday is the best day for you. No MM. Not near a weekend. Go to DLR on July 7.
