Best method to rewrite Request and Response URI, preserve URI data, and with no client redirect

Brad, I apologize for the delay. We were not able to test this as the systems were Production. We bought a set of Dev-QA F5 VEs, deployed them, and are testing this solution. Unfortunately things aren't working as we'd hoped. In doing packet captures the request is getting re-written but sending a 303 back to the requesting application server which causes an error. Let me see if I can update what I see as the task our Dev/Middleware team is trying to accomplish.

1. Client Browser Request comes in https://www.site1.com/123 (SSO site).
2. Backend SSO application server authenticates and makes a call to 123 (IAC) application VIP using https://www.site2.com/123
3. The URI of /123 needs to be re-written prior to going to the IAC application server to /321 with no 303 or redirect message sent to the requesting application server. Doing so causing a failure.
4. Once the IAC app processes the request /321 it will respond with /321/blah/data and will need to be re-written to /123/blah/data prior going to the requesting application server (SSO) and then back to the client. We had an F5 training class with the guy that wrote the training modules for iRules, but unfortunately we did not get to take a look at this prior to his early flight departure :| Hopefully you or someone on here can provide some guidance. Is this something that is going to require an OnDemand iRule? One final note is we are running v11.6 and tried a rewrite profile; however it also sends 303.