There is no relationship between cookie processing, and certificate extensions. The path length constraint (in the
Basic Constraint Extension of a CA certificate) enforces a limit on the length of the certificate path beyond that CA; with a path length constraint equal to 1, that CA certificate is good for issuing non-CA certificates (e.g. a certificate for a SSL server), but not for issuing other CA certificates.
Cookies occur at the HTTP level, when the SSL tunnel has been successfully established. There is no HTTP, hence no cookie at all, if something goes wrong with the SSL tunnel; conversely, when cookies come into action, everything already went fine with the SSL handshake, in particular the validation of the server certificate and the processing of certificate extensions that this entails.