

Not sure if I fully follow what you want. But for my use case, I have something similar with cookie auto delete extension and container tabs.
The extension (not firefox) is set to auto delete everything by default. And then I can add a website-container pair to the extension’s allow list and it just works
The only difference I can think of is that I don’t bother with individual cookies, rather within the container, all the cookies from the particular website are allowed
In the settings for CAD, there’s a checkbox to enable support for container tabs. Have you selected that? It’s under extension options