It’s likely to be pretty claggy in places so personally (and at my level of fitness) I wouldn’t recommend going for a one-day each way attempt especially given the shortish days we still have.
My folks rode it about a decade ago and they managed to train back to Southampton (where I lived at the time) without a problem coming back from Eastbourne I think, which involved cutting off a little bit at the end. It’s a pretty slow service (stops just about every station) but workable.
The train company you need to check out from Eastbourne/Brighton end is Southern.
When you get to Southampton (Central or Parkway) you can then easily get a South West Trains service up to Winchester as that’s the main line into London.
You probably want to make sure you have some means of getting the worst of the mud off your bike and you (or at least covered up) before you get on the train.
One other thing I would suggest is NOT TO leave your car on top of Cheesefoot Head for two days.
HTH
Kev