Well, DigiKey has a relay (several, actually) with a 12V, 960 ohm coil, that'd be a time constant of 4.512 S, which would hold up for a glitch, but that's about it.
Now, if you really insist on letting a cap discharge, just amplify it. :-) Use a MOSFET, although you'd probably want to use another transistor for a little positive feedback so you'd get a snap action - I don't know if you'd want your MOSFET to be in the linear region for 2-3 minutes. :-)
Cheers! Rich