Cisco VPN Client interrupts connectivity to my LDAP server

It sounds like the Cisco VPN Client has been configured to block access to your local network and only provide network connectivity to the internal VPN network. This is normally done for security reasons to prevent bridging the networks together. As part of this it has set your DNS settings for the VPN to be the VPN LAN ones so that you can resolve DNS names of your company network hence your problem.

There will be an option to "Allow Local LAN access" in the VPN settings but that is normally administratively turned off. Here is a link showing you How to turn on Allow Local LAN Access but your VPN admins have probably locked that down to prevent it.

It may be a question of certificates if you're using ldaps (ssl/tls) to connect. The bind done with a reverse ptr lookup address will fail because it's not the address given in the x509 certificate. There's an option you can put in your ldap.conf/slapd.conf: reverse-lookup off

There and again it could easily be something else but worth checking I think