Here is what you can do without any theme code edits.
Add a new link to your menu, call it Chat and make it point to #chat (address is important).
Then add a “Custom liquid” section into your Footer section group and paste this code:
The code will listen to click on any link like … and will open/close the chat app.
Preview in my test store: https://qf4vm29deon22n7j-23104437.shopifypreview.com – click “Chat” in main menu.
Not quite the button you’ve asked for, but otherwise would need to modify the theme code which I currently have no access to.
OR, can decorate this link with CSS…
