Projects that require 2 power supply voltages (Raspberry Pi)
I tried using a 12V to 5V buck converter board, connecting the 12V to the Picobuck and the 5V to the Pi.


That is the basic approach I would take too.


  This is fine until I switch on the LED, which causes the Pi to either reboot or shutdown. I presume there is a voltage drop, or not enough current to keep the Pi running stable?


My guess is what is happening is when the picobuck turns on it draws a spike of power that glitches the system power supply.

Unfortunately without an oscilloscope it's difficult to debug this but I would start by adding large capacitors to both the 5V and the 12V rails.