I want to use Register Shutdown Function to track Fatal Errors, my question is, where's the best place to call this so it hits both WordPress, Plugin, and Theme Fatal Errors?
WordPress core register a shutdown function on its own (source) the function
and little more (source).
So, if you want to register a shutdown function in WordPress way just add your function on 'shutdown' hook:
Where to put this code is not very important, a plugin, a mu plugin, functions.php, it doesn't matter, it will run always on shutdown...
Of course register it as soon as possible allows you to catch early errors, so using a mu plugin you can catch errors that happen on plugins init.