Delayed button press
The voltage on the mic line is the bias voltage for the electret mic element.  I think it comes through a 2.2K or 2.7K resistor.  There is not enough current available there to power anything.  You're only option is an external power source.  One lithium coin cell should last a long time.

Separate from that, your two circuit options are a small batch of CMOS chips or one of the ultra-low-power microcontrollers (PIC, AVR, etc.).  If the uC doesn't not have an open-drain option for a GPIO pin, you'll need one external FET like a 2N7000.