Which cookie should I keep to prevent new logins at every browser start?

Thanks for doing that troubleshooting.

One difference between the 1Password extension and most other extensions you may have is that 1Password requires some additional data for it's end to end encryption and the extension stores it in local storage. Clearing local storage data would lead to the behaviour you described above as not everything can be stored in cookies. Could you double check that your Cookie.app configuration is set to leave local storage untouched for the web extension? Above you mentioned that you found safari-web-extension_<UUID>_0.localstorage in Cookie.app, that sounds like the right thing to exclude from cleanup. I installed a demo of Cookie.app to have a quick peek at my own extension data. I primarily use Chrome and Firefox myself so the presentation is different but the principal is the same. I do see that Cookie.app is able to find my local storage and it correctly identifies it as extension data, is it doing the same for you as well?