Removal of Client Side F5 Persistence Pool Cookie

This is exactly how Cookie Persistence method works and if you remove the cookie you break the functionality. There are alternative cookie persistence methods you might consider. Check them below:

Overview of cookie persistence (f5.com)

In addition to it, I recommend you to enable encryption on your current cookie persistence method. This will prevent  malicious users to revert the value of the cookie into IP address of your pool member.