The way the plugin is written, it adds the tab in the footer of the site, then uses absolute positioning to stick it to the side of the browser. It order to pull this off the way you need it to, the tab should be placed somewhere inside the main content container, then with relative positioning, you could adjust it to the side of your content container. My plugin does not support this solution.
You may want to just copy the tag and CSS the plugin generates and stick it within your child theme and adjust the CSS accordingly. I’m not sure how that would work on mobile without testing it.
If you get it working this way, I would like to see it. That would be pretty cool.