I would want to "electrically" insert | or \right| when typing | or \left| in AucTeX and can't quite figure out how to add characters to AucTex-electric. A list of already defined completions can be found here. Can anyone help me with this issue?

(add-hook 'LaTeX-mode-hook

Maybe, (setq LaTeX-electric-left-right-brace t) is not really necessary since you have it already customized or hooked in. Nevertheless, it does not hurt if you keep it.