The circuit below does what you want. It
At all other times the relay coils draw no current.
This is a "cut down" version of my circuit from this answer to "Power Latching Relay off/on in response to 12V ignition and 1 second ground pulse".
If there is a single ignition feed input then an overall power supply capacitor - charged via a diode (not shown) needs to provide enough energy to hold the IC on and to provide relay delatching energy. If an unswitched 12V feed is available then the circuit can be powered from that. After the turn-off action current drain is around 1 microamp.
I have left an additional a 1 second input pulse option to show how the relay may if desired follow a shorter pulse. The two inputs are essentially identical.
A "better" solution may be to use an Arduino or similar as more flexibility is possible if the spec "evolves", as happens. But, this should do the specified task.
Inverters are a package of hex Schmitt trigger inverters.
MUST be Schmitt trigger. (CD40106, 74c14, ...)
There are two spare inverters which can be used in various ways if desired.
D1 accepts 12V positive on ignition
D2 (optional input) accepts a 12V positive going alarm pulse.
R1C1 and R3C3 plus following inverters provide 1 second delays.
A negative going edge pulls the cap low and the resistor then charges it high.
When the alarm pulse activates C1R1 the output pulse lasts the lesser of the alarm pulse or C1Ra timeout. As the set/reset relay only needs a 10 ms pulse 'this is not a problem".
R1C1 & R3C3 both provide ~~= 100 ms time constant but values vary by a factor of 10 as examples of how different values can be used.