This extension can be made without a content script and with many fewer permissions. You should look into https://developer.chrome.com/extensions/commands to see how to properly integrate keyboard shortcuts, and https://developer.chrome.com/apps/app_window for the floating UI.

Unfortunately, Chrome extensions are not allowed to use chrome.app.window.create (which you will need to make the floating UI without modifying the content page), so you will need to convert your extension into an app to use this API.