Buy a fence panel the same as the existing ones but perhaps only the height of the bins (4ft big enough?) Attach on hinges to a vertical piece of wood which you screw into existing six foot fence at one side (top and bottom). Your new section of fence can them be swung out to open it and gets bins out when required but the rest of the time the bins will be mostly hidden behind it. And you can still easily open them to put rubbish in without having to lift any wooden cover or anything.
You could also paint the kids of the bins brown to make them blend in. The bodies of the bins will still be the different colours so it’s easy to see which is which.
If you have a car, measure how big a panel you can fit in it prior to buying, if you want to save on them having to deliver it.