You can get a hose connection for the water butt, allowing you to feed a pressure washer with it.
This. I actually have a 200 litre water butt suspended in the rafters of my garage. Rainwater gets pumped up there until it's full, then it gravity feeds a Karcher nicely. If you live in a hard water area, you'll notice much less residue on the paintwork after washing, but you still need to dry off a dark car with a chamois for best results. No way round that, really, although you might find it easier to blow the water off with an airline, perhaps?
Alternatively, if you have a water softener, use softened water to wash and rinse your car.
The water leaves a residue on the paint as it dries, so the trick is to reduce the residue in the water in the first place (soft water or rain water), then remove as much of the water from the paintwork as you can before it's had a chance to dry.