It is one day and is a scheduled event every year. People ought to be able to plan around it. If they can't be bothered to think and plan, they are to blame. It is the same with people who whinge when their local street is closed for a cycle event, as though it is impossible to park their car somewhere else for a couple of hours, or get their shopping before the road closure. People just can't be bothered to think, this is a common problem in the SE.
You don't need to stop the event, but if it is becoming unmanagable, perhaps it may be time to consider moving it somewhere which does have the room to host it. If that is not possible, scale it down.
The public transport is disrupted because it is easy to do. There are only three ways into Lewes by rail, so trivial to block those three routes. It would be far more difficult to block all possible routes by road, which would be difficult to do without causing transport problems for people just trying to get somewhere other than Lewes, plus you'd have to block all the public rights of way, which would be pretty much impossible.
