I discovered about an hour ago, from the desk at Temple Meads, that advance booking is also required for the Cardiff-Portsmouth trains, although in the words of the staff member who served me "They're a bit more flexible on those trains". Just as well I did ask as I'd been assuming that if I didn't get the 0700 Paddington train (to Bath) I could always get the 0722. But no. (So I booked on to the 0722 for the sake of being a bit longer over breakfast.
) I was also told, contrary to what has been mentioned before, that it's not possible to book a bike space on multiple trains "just in case".