Besides the batteries and the converter, the problem could simply be a bad connection or wire (usually where it connects), causing a high impedance connection between the lights and the 12V power source. Make sure the cables at the battery are all in good shape and the ends clean where they connect, and the ground connections are likewise sound where they connect to the frame.
This is fairly easy to check, actually, beyond a visual inspection. If you measure the voltage at the battery terminals themselves, and it stays (nearly) the same while the lights are changing in brightness significantly, then clearly there is a difference in voltage between the battery and the rest of the system, indicative of a poor connection somewhere between them.
