I have tested the Lite Speed plugin with Private Cache for logged in users on my staging website. So far, it works without excluding the Simple Membership cookies. If you want to try it at your end, please let me know. I can send you an export of my settings of the Lite Speed plugin. You only need to install the plugin, import the setting, and you are good to go!
Why is it advised to exclude the cookies from caching? To prevent protected pages to be cached and shown to non-logged visitors? (otherwise, the protected pages became Public by means of cache. It used to happen with Fastest cache).