Inserting HTML on Account Dropdown via JS

Hey there! I’m wondering how I can insert html on the my account dropdown via JS (would need to be called when the account dropdown is loaded). I use api.onPageChange((url) => { for a few things but I need a different event to trigger off of for this.

Would prefer to do this with JS and not a plugin.
https://gyazo.com/3ccd84dcdda1d012e7f2118c6f77cdd7