I don't have much experience at all, but don't worry this is not another Darwin moment and I do at least know not to work on a live circuit!!!
I didn't quite understand what you said though. Let me try and summarise what I think you are recommending.
Existing light fixture: take the black and white from that and join it with the black wired from the billiard table lights and leave a pigtail connection that then goes in to the brass connector on the pull switch.
do the same for the white, only connect that to the silver connector on the pull switch.
I thought that wouldn't work because I'm not interrupting the power supply to the billiard lights with the switch?
I'm sorry, I'm sure you are rolling your eyes, I do appreciate your patience. I don't mean to worry you but I wired up an entire room in my basement with lights, sockets, and even a pool pump only 3 years ago but they were all new circuits and properly inspected by the authorities. I can't understand why I am so stumped by this one...age....is a cruel cruel trick.