I think I need more info about how to control the external circuit.
Now, with using also an external relay, the timer can be trigger by a short contact of the door switch, (this contact = normally open) stay on power for the programmed period and automatically disconnect from the power supply end of the time.
The most important thing missing. What about the external contact?
There are some possibilities you may wish to implement:
1. NC normal closed contact ( closed for the power OFF), stay closed for a time period after we apply power to the timer, and then to be disconnected only for a short time (how long you need this ? ) after the programmed timer period has expired, later back to be closed
2. NC normal closed contact (closed for the power OFF), disconnected for the whole period timer ON, but reconnected when timer OFF
3. NO normal open contact, kept open for a time period after we apply power to the timer, and then to be connected only after the programmed timer period expired, (but for a short time – so how long you need this ? ), and later back to be open.
4. NO normal open contact (open for the power OFF), connected for the whole timer period ON, but reopened when timer OFF
Not all the options can be prepared with the actual timer, so need to choose what’s the timer action you expect.