Thanks for all the follow up responses. Will call the train company to see if it is do-able.
I'm planning East to West based on the wind forecast (as also suggested above). 5.20am train from Winchester gets me to Eastbourne after 8am. That fits within the rules for going in and out of Clapham Junction, but I wasn't aware of the Brighton restrictions. Ugh!
With the clocks changing this weekend, I'm hoping that I'll have just about enough daylight. If I can start in Eastbourne at 8.30am, that would give me 11 hours until sunset, and hopefully I could just then use a light for the last section if I'm taking longer.
Now if only I can get these darned trains to work out. Otherwise I'll be in serious brownie point debt to my pregnant wife if I ask her for a lift from Winchester to Eastbourne at 6am!