As far as I can see the Plasma LEDs are standard APA102s, which work over the SPI bus. One of the Plasma buttons product pages says
As well as using Plasma buttons with Picade and Picade X HAT, you can connect them straight to GPIO on your Pi, or other microcontroller. This makes them ideal for DIY arcade builds and even building things like your own DIY midi controller
So that seems about right. I’d take a wild guess that the Plasma CLK pin should go to BCM11/SCLK and DAT to BCM10/MOSI, but that’s just a guess.
Ah, I was wrong it seems. The Plasma buttons tutorial says:
You’ll see the pins marked on the Plasma PCB, 5V, GND, CLK, and DATA. Following the wires from the Plasma PCB connector to the other end with the male pins on, connect the 5V and GND pins to the 5V and GND (it doesn’t matter which) terminals on the EXPAND side of the Hack Header on Picade X HAT. Connect the CLK wire to the terminal marked BTN8 on the Hack Header, and the DATA wire to the terminal marked BTN7 on the Hack header.
Pinout.xyz shows that BTN8 is connected to BCM14 and BTN7 to BCM15, which is the UART bus. Not sure how that works, but that’s apparently how the Pimoroni software has it set up.