Thanks guys.
I don't always really trust Optimates, though, but I think keefer's suggestion of charging it and leaving it off the bike is a sensible one.
I've just taken it off and put it on charge. Although the Optimate only shows "half-discharged", the meter only reads 9.7V across the terminals. That sounds very low to me; I reckon it's clapped.
I'll see how it looks in the morning, then leave it until tomorrow evening and see how it does.