Demolition Derby Track
If the FBC and LCH are built slightly overlapping, as shown in the picture, then maybe they could cross connect the rings to allow particles to travel in figure eights, in both directions of course (courtesy of counter-rotating beams), and then hold a super destruction derby.
OK, OK, I know that the magnetic fields of the LCH wouldn't contain the more powerful beams of the FBC.
But still, as someone who used to squirt lighter fluid onto plastic model u-control planes for more realistic crash scenes (Will the burning plastic pilot escape the wreck? Of course not!), I immediately start thinking of how one could created a really big bang.