Right, we don't know your network environment. There probably is a way around it. SNAT makes things easy, but quite likely you don't need it if you have control and can plan your environment carefully.
If you don't want to change all your routing tables to actually see the client's ip, then you probably should just use timestamps. Ensure your clock is the same and you can probably figure out the user from the logs. You could also try to embed something in the packets sent to the DC, but you'd also have to configure the DC to strip and log that information. You could do the F5 part easily with irules, but the rest would be up to your server configuration team.
I'm not sure of what kind of "troubleshooting" they "can't do", but it seems like you might have to work in concert with them to identify the clients as the ip alone is not enough.