The delayed output doesn't have the current drive capability of the initial alarm pulse.
The starter relay requires far more current than a 555 can deliver.
hi,
The current drain is quite small when the relay is not energised, I would always connect a project to a car battery via a fuse, say a 2A fuse, in the +V line.
Fuse in series with the D2 diode, close to the battery end of the +V wire.