How can you delete a website's stored data but NOT its cookies?

I have signed up to a website that requires login stored in one or more small cookies and also stores a large amount of data locally to carry out processing. My Firefox desktop 153.3 ESR is set to delete all site data on exiting unless listed as an exception, which is what I want for most sites for privacy and security. For most of the exceptions that are not deleted, keeping local site data as well as cookies is not an issue and it is essential to keep local data for some. However, for this one site, I want to KEEP its cookies so I remain logged in but DELETE the site data automatically on exiting Firefox. I cannot see any setting for individual websites to handle cookies and local data differently, while changing the overall option for what is cleared would also delete the local data for other sites that is essential to keep.

Is there any way of configuring Firefox to delete a website's local data on exit while retaining its cookies without deleting otherexception websites' local data? If not, providing granular control of what is kept and deleted for website exceptions to the general clear on close would be a very useful feature to add. Thank you.