Currently the strategy is to use some kind of pin/socket pair on the wiring, and leave the cable long enough to allow for the rotation, or counter-wrapping the wire around the carriage post when closing the shells.

New connector idea: TRRS (tip, ring, ring, sleeve) connectors, aka headphone plugs, between the LED shells and the carriage.  It slides into place, cushioned by a silicone plug at the bottom of the socket, as the carriage screws into the shell.  This would weaken the carriage screw though, so maybe replace it with a short hollow threaded rod and put a short captive nut in the shell somehow.  This may also allow for another LED to be placed underneath the connector for maximum coverage.