While a capacitor across the traces that power the LEDs will help, the voltage range over which the LEDs light is small, so the capacitor can't deliver power very long before the LEDs dim and then go out. I don't know anything about the Walthers lighting units, but if the rectified diodes are connected to the leads from the rails, then you have about 12V DC after them somewhere on the board, and one or more resistors setting the current to the LEDs, which operate at about 3 volts. If you can attach the capacitor to that higher DC voltage PCB lands, then it can power the LEDs with little change in brightness as the capacitor discharges a few volts. I think you'd like that better.
The disadvantage is you need a capacitor rated for 15 volts or more. Higher voltage caps are bigger, but the capacitance (microFarads) can be much lower to store the same amount of power.Actually, their volume is about the same for the same energy storage capacity, which makes sense. A capacitor rated for at least 100 uF (microFarads) and 16 or 25 volts should help a lot this way, where directly across the LED power lands, you'll want 1,000 uF or more, but only about 6.3 volts. An even lower voltage rating would be better. of you can find them, but they should be rated for at least 4 volts, and I'm not sure anybody makes caps with those ratings.
