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This is a problem with all extensions that use IndexedDB for storage. Firefox blocks access to IndexedDB storage in private windows, even by extensions.
If you like automatic private browsing more than you like convenience, you can create a new folder named chrome in your Firefox profile folder, and copy/paste your style rules to a new file named userContent.css which Firefox will read at startup.
More info: http://kb.mozillazine.org/UserContent.css
Optionally, you can create a file named userChrome.css -- chrome refers to the user interface parts of Firefox, content refers to web pages. I created a website for that file which shows where to create the chrome folder, so that might be helpful for general orientation. https://www.userchrome.org/
Or you can switch to Stylus, which works in private mode. As well as inside mixed private/standard/container tabs.