Nimbostratus
Cirrostratus
Are you using a session cookie (ie no timeout) on the persistence profile? If so, the BIG-IP shouldn't need to set the persistence cookie on every response. The client should continue to present the session cookie in every request to the VIP regardless of whether it's been set in every response. The reason you'd want to set the cookie in every response is if there was an expiration time on the cookie. Re-setting the cookie on each request would tell the client to continue sending the cookie in subsequent requests until the new timeout expired.
With that said, do you have a specific reason for thinking that setting the cookie on every response will fix the persistence failures? If so, it's simple enough to check for the cookie in requests and set it again in the response. The persistence profile will dictate that the cookie is set on the first response to a request without the cookie. If you don't have a specific reason, I'd suggest looking at other potential configuration and/or pool issues first. For instance, do you have a OneConnect profile enabled on the virtual server? This will ensure that each request on a TCP connection will be parsed for persistence decisions even if the request is going to the same pool as the previous request on the same TCP connection. Also, do you see the pool members being marked down? This will cause persistence failures because the original node can't be used for subsequent requests.
Aaron