Southern don't do bike reservations, but a MkI Winning Smile and a touch of being sensible should see you OK. It's officially two bikes per set of four carriages, but I've seen half a dozen.
Well maybe if it's not a problem if you don't get on, and you have to catch the next train, but I'm not sure I'd chance it - especially if you're also relying on the train not being late and not getting lost on your way across Portsmouth.