OP why don't you stay at an Air BnB on the outskirts- Watford for example as a pp said, or somewhere in South London- that's near a tube or overground station, and then just use public transport. Just make sure you get a place with street parking so you can leave the van at the house/flat.
Download City Mapper app on phone to help you get around.
Where do you live/where are you coming from? Might make it easier to suggest areas to stay.
Loads of London families rent out their homes in the summer and its much better value than a hotel for a weekend, especially for a family of six.
Depending on how many you need to pay for, just use contact less cards, one for each adult/older child, and tap in/out. No need for oysters.
Museums as you say are free. If you're at an Air BnB you can get a supermarket delivery to arrive so you have enough food for the weekend and can take packed lunches/snacks out with you in the day and then eat at home instead of eating out.
Loads of wonderful parks and free things to do in this amazing city, you just need to know where to look. All the big museums are bunched quite close together which makes planning a day out reasonably straightforward.
With some careful planning London doesn't have to cost the earth . London is great, especially in the summer, and as you say the covid time slot system means many places aren't as busy/stressful as they usually are.