Slowing Down Capacitor Circuit discharge rate
I think adding a resistor will defeat your purpose. The relay's coil serves as a resistor, shunting off the capacitor's charge. Add a resistor in series with the coil and you may not have enough current to activate the relay. Add a resistor in series with the capacitor (parallel to the coil) and your capacitor may not build up enough charge to do much of anything. Add a resistor in parallel with both the coil and the capacitor, and you're just pulling more current (besides discharging your capacitor even faster).

The coil here is equivalent to a resistor in a simple RC network. Its resistance is 12V / .15A = about 80 ohms. Through experimentation (there's no good way to do this part through calculation), determine how low the voltage can go without dropping out the relay. Now from THAT you can calculate your timebase - how long does it take a 1000uf capacitor to drain from 12V down to (the dropout voltage) through an 80-ohm resistor? Not very long.

If you add too large a capacitor, though, CHARGING it through your indicator may damage the indicator, the blinker, or the switch... and will almost certainly blow the fuse.

Try powering the relay from a 555 timer, preferably with a driver transistor. That takes a MUCH smaller capacitor to accomplish the same goal, uses far less current, and gives you some more soldering experience.    8)