Cookie insert persistence seems like a good solution here as well. You can use this format:
persist cookie insert "cookie_name" 3600
This will trigger the use of a persistence cookie named cookie_name. If you want to use a session cookie that the client should keep for the duration of the browser being open, you can remove the timeout from the end. Else, LTM will set a time-expired cookie valid for one hour.
To test multiple HTTP requests on the same TCP connection manually, you can use nc (netcat) from the command line of LTM itself (nc 1.1.1.1 80) and then type/paste in the HTTP headers you want to send for each request.
Aaron