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I've done part of the South Downs before and I don't remember it being too bad (I did crash in a rut though so maybe my memory of it is worse than I thought).
I've not looked closely at the route but if we can get around in hotels then it'd be nice not taking camping gear. Then again, we have all the camping gear so that's also an option. Wanted to use it more of a shakedown ride before Seven Serpents.
Avoid the South Downs Way for the next couple of hundred years and you'll be good.
For the rest of it, I'd wait until late May at least given the current ground conditions and forecast if you are wild camping