A simple (less general) solution is this:
Create a global singleton (e.g. GlobalState.gd); could look something like this:
extends Node
var selected_character = 0
var selected_difficulity = 0
# other global config stuff if you like
Then in the menu controls do stuff like when the user taps "ui_right":
GlobalState.selected_character += 1
For the preview in the menu and for loading in-game you can just read the selected character like this:
GlobalState.selected_character
Also don't forget to register your global singleton as such in the project settings.