As an alternative to a second pickup, or in addition to a second pickup, you can put a bridge rectifier and a capacitor into the caboose. One ~ terminal of the bridge goes to the frame, the other ~ terminal to the pickup(s). The + and - terminals go to the lamp and to the + and - terminals of an electrolytic capacitor (5 or 10 millifarads, 25 volts minimum) in parallel.
With the rectifier (with or without the capacitor), there is also a way to connect the second pickup so that AC current cannot flow between the pickups, by using a separate rectifier module for the second pickup and connecting the + and - terminals of the two rectifiers together. (This can also be done with one bridge-rectifier module and 2 diodes, or with no rectifier modules and 6 diodes.)
NB: A millifarad is 1000 microfarads.