OK, we're going to try it. Southeastern into Victoria can't tell me which train type it will be, though in all probability it's the suburban sort (465?) without any particularly good storage areas. They tell me, though, that as far as they're concerned on Saturdays a bike is a bike and we are welcome to travel with the tandem if we can get it on board. I think it will fit on and we can just half-block an exit door.
Then a 377 with Southern to Portsmouth. It won't go on the Island Line railway so I might just get myself a ticket to Ryde (and I don't care).