The $llive_site in configuration.php should be blank. Only specify it if you absolutely understand the purpose of it and if you specifically have a need for it.

Regarding your issue it's only going to be solved once the mechanism for triggering session login is found. I at this time do not know what triggers it, but I'm pretty confident it's Joomlas user system plugin. I however don't recall when. If it's after initialization then your issue can't be solved unless Joomla fixes it as it's not triggering early enough. I believe it would need to trigger before routing as the Joomla menu item access checks are all done after routing.

I recommend leaving $live_site blank and and Rewrite URLs disabled since it solves your issue for now unless you're having issues with users reaching Joomlas login page instead of CBs.