FLC container is injected in code without location choice for the developer

Problem/Motivation

I have installed the FluidUI module on my D9 website. It works fine, can use all the settings to change look and feel of the website to accommodate all people with accessibility issues.

However, it injects itself into the page code, rather than giving me options (a custom block for example?) where I'd like to place it. As a result I can only have the preferences tab hanging from there very top of the browser, where it overlaps part of the menu etcetera.

I would prefer if I could choose where to place the tab, for example middle left or right of the browser window, with an icon, or below the header so it overhangs the hero image.

Any help on how to achieve this would be amazing!

Proposed resolution

Add options to the module configuration (currently only 'show on admin' and 'hide on these pages') for where to place the tab, use icon and/or text, supply a custom block that the administrator can her/himself place on the page in different locations.

Alternatively, where in the module code can I tweak where the code gets injected (different page region?)