since what you need is just an impulse, hope the attached circuit may work. it may give you a suden on and off in a fraction of second and this delay is decided by mechanical response delay of the relays.
this is how it will work.
when the circuit get 110V relay R1 picks since it has a path via normally closed (NC) contact of relay R2
when R1 pick R2 also get power via normally open contact (NO) of R1, then R2 will latch since it has another path via its own NO contact, same moment R1 will drop and remain the same.
until 110 v is removed R2 will rmain picked.
R1 just got on and then off and remains.
so the second NO contact of R1 will give you a momentary impulse when you connect your 12v circuit via it. it may be in a fraction of second but of cause much enough to trigger your lock
you dont want to go for any expensive electronic time delays for your need