Here's a portion of the datasheet:

Does that mean the coil must have 5V of voltage minimum to work? If so, it would explain why putting a resistor in makes it stop working -- since there will be a voltage drop on the resistor?

For simplicity, according to my calculations, with a 1000uF capacitor I would need a 200 Ohm resistor to get the time constant of 0.2 second. Since it takes 5 such constants to charge a capacitor fully, I will get my 1-second cycles. I might be off, but it's close. The problem however is that this relay stops working with much smaller value resistors.

Any ideas are welcome!

Thanks!!