Would I need one with a fused output?

Yes. It would share (would have to share) the fuse with its own sockets.

If I make the triple socket the first spur, wouldn't the fuse in it
protect the wire to the first spur

Yes.

against an overload from the triple itself and the double spurred from
it?

Only if it had a fused output. If you spur from its input, then no.

Effectively I would be using the fuse in the triple to do the same
job as an FCU. What am I missing here?

Nothing, except that the accessory might not exist.

Christian.