I have now reached the point where further diagnostic testing is best done on your test site. So far, I have been unable to see, on my test system, any of the issues that you are experiencing. The real "eye opener" for me though is that, when I look at your test system with my web browser, I have all the same problems that you do, but, and here is the key point: the Cookie is always there, in my browser, with the correct value in it.
At this point, I will need not only an ID and password for WordPress Admin on your test system. But I will also need FTP access into the WordPress installation directory and all subdirectories, with both read and write permissions. I use FileZilla so I can use whatever FTP access you prefer: SFTP, FTP, or implicit or explicit FTP over TLS. This will allow me to edit my plugin files quickly and easily for debugging purposes, typically adding a var_dump() in relevant places.
I am sorry that this has taken so long, but things have been busy for me, including a high volume of plugin support issues. And, of course, this is a very odd problem, with lots of possibilities to explore.
Thanks again for your patience!