UPDATE ON THE PROBLEM:
I think you are mixing up things or you should go into more detail about your setup/problem.
PHP's session path is the location where session data is stored on your server, not the client. See the documentation: https://secure.php.net/manual/en/session.configuration.php#ini.session.save-path
You can move these files and replace/keep in case of collisions how you see fit. This is pretty much only restricted by read/write-permissions you have when accessing/moving stuff and your webserver-user (e.g. apache or nginx) or php-user has for reading/writing them from/to the new location.
If by "PHPSESSID in their browser" you mean the session id is part of your urls, that is a different PHP-setting, that should be disabled anyway, see notice in the documentation: https://secure.php.net/manual/en/session.configuration.php#ini.session.use-trans-sid
edit based on your updated question:
There already is a nice JS-based solution for expiring the old cookie. I would go with that. if you can't just do that, you could do a redirect to /cv have a php-script there that reads the cookie and stores the data somewhere (a database for example based on the user_id) and expire the cookie. Then you can redirect to the old page, look for the "/"-cookie and restore the data. It's a very ugly hack, but I don't think you can get the cookie for each path in PHP, since it's server side and based on the session id provided by the client (but I might be wrong).