Capacitor to discharge upon switch open
We can simplify your requirement specifications with a simple timing diagram.

           __                    __________             __    __
Button ___|  |__________________|          |___________|  |__|  |_________
              _____                         _____          __    _____
Output ______|     |_______________________|     |________|  |__|     |___


Figure 1. Timing diagram. This shows us that the load is off while the button is pressed and turns on for a preset time when the button is release unless the button is pressed again (as shown on the right of the diagram).

The challenge, as you have discovered, is to do this with a single-pole, single-throw (SPST) switch or simple push-button. A common solution to the limits of a switch - it might not have adequate current or voltage rating or may have the wrong contact arrangement - is to use a relay.







simulate this circuit &ndash; Schematic created using CircuitLab

Figure 2. Using a relay to convert an SPST button into a SPDT (single-pole, double-throw) device.

How it works:


SW1 you have already.
RLY1 is a 12 V relay with a changeover contact. You should be able to find one in a car parts shop or at the scrapyard.
In the normal position C1 is discharged into the load.
When SW1 is pressed the relay is energised and C1 is charged via R1 which limits the current into C1. (When C1 is completely discharged it appears as a short-circuit at the moment of switch-on.) Note that the load is completely disconnected during the charging cycle.
When SW1 is released C1 discharges through the load.
Note also that if SW1 is pressed again before C1 is completely discharged that the load will be switched off as shown on the right side of the timing diagram.
D1 protects the contacts of SW1 from sparking that would be caused by the inductance of RLY1 during releasing of the switch.


One big problem with this circuit is that the voltage will fall as the capacitor discharges and in the case of a bulb, for example, it will flash brightly and then fade away over a period of time determined by the lamp resistance and the capacitor value.