After some tests with a quick bike pannier prototype, I tried a few different designs and then built a pair of custom bike panniers (from the same surplus marine upholstery I'd found for cheap). They were sized to fit my 2-person tent, foam pad, and sleeping bag inside one of them (tents are usually too tall to fit in a pannier), and to be fully waterproof. This was tested in the field by leaving them full of my clothes and maps in a torrential downpour overnight.

The design and construction process, in a few photos:

They treated me well on a solo 600-mile bike camping trip through The Netherlands and Belgium: