The persistence value isn't generated in a response. The persistence value is part of all connections to the server, including the very first one.
Also, you're saying the timeout for the persistence value is a timeout on when it was added? I thought it was a timeout on when it was last accessed, so just doing the
persist uie $uid
should restart it.
Not sure about the applying to virtual vs persistence profile. I believe it's applied to the profile, but I'm not sure (I'm not the one doing the configuration, just the one writing the iRules).