I commute to Victoria and London Bridge from Eastbourne sometimes, there’s lots of direct trains, if you travelling on a weekend you should be fine with your bike, in fact there’s dedivated bike carriages, only folding bikes are allowed during the rush hour on week days though. Then as mentioned I’d ride to you next connection.
If you’re in Eastbourne for the weekend and have the energy then try and head over to Friston Forest, lots of hidden singletrack and a marked red route, there’s also the famous seven sisters white cliffs at Cuckmere Haven.
Have fun, tempted myself.