the relay is a oscillator that discharges when a capacitor reaches a certain value, by doing the conversions you have done, you have decreased the resistance of the circuit and allowed a greater current to flow and charge that cap faster. You COULD add a resistor so that the circuit sees the original factory resistance but you will decrease the amount of current going to each bulb and they will not be as bright.
If you wanted to be abstract about the way the circuit works, you could say you needed a switching amplifier for each bulb, maybe made with a transistor. Then you could use the factory lead as a control lead and the amplifier as the source for the bulb. Note: This is the kind of thing I would build in the lab at school. I would probably use a few MOS transistors and a PCB to solder the circuit together. Maybe I will design something like this and post it here if people are interested, although im sure there is a better and more efficient solution to the problem.