I would have gone this way:
Bristol Temple Meads - Southampton Central. Which you can book ADV tickets for, but the off peak returns are cheap enough I find

Then Take a Southern Service to Brighton, they run every hour, normally at 33 minutes past
IIRC▸ . Normally these depart from the opposite platform on the same island. So stairs aren't an issue, IIRC there is lifts there so that shouldn't be a problem. But if you want to guarentee the train being on the same platform, split the tickets are fareham instead. (Something else to consider, with thanks to Brucey for this, is the split at Havant trick, which is slightly cheaper than
BRI» - Fareham).
When you arrive at Brighton, this is terminus station with total step free access, bar the odd pavement or two...

Pick up a direct service to Eastbourne, and your there!
Completely avoids LU and mostly step free
