Split power between LEDs and PCB
I'm designing a system that runs WS2182B signals down RS422 lines to LED strips powered elsewhere.

that seems like a good idea.

but I can't figure out what connector(s) I would use.

If the led strip has a connector I would use that, if not maybe make a PCB with notches in the side and solder that to the strip.
solder the power wires to the strip too