If the Discord app is listening for system-wide keyboard shortcuts, then it will be in conflict with other apps that also do the same, such as AutoControl.
In this situation, an app may or may not see a given keystroke depending on which of the 2 apps received the keystroke first.
In order to work around this cat-and-mouse problem, you have to configure Discord to accept a second shortcut for the same action as F12; for example CTRL+F12.
Then, in AutoControl you must do this:
This way, when Chrome is focused, F12 will be overridden and Discord will see a CTRL+F12 combination.
When Chrome is not focused, F12 will not be overridden and therefore Discord will see the keystroke and react to it.
If the Discord app doesn't allow you to specify 2 keyboard shortcuts for the same action, then you can override F12 at all times (i.e. when Chrome is focused and unfocused) as follows:
And then configure Discord to use only CTRL+F12. This way, pressing F12 will always trigger a CTRL+F12 combination which Discord will detect.