It sounds like the servers are configured via host headers to only answer for their specific host name. If you could modify this to the hostname which resolves to the virtual server IP address, then you wouldn't need to make any changes on LTM. So you could have stores.qa.stores.com resolve to the external IP 65.x.y.z. That public IP would be NAT'd to the virtual server IP, 10.10.10.1. Each of the servers would be configured to answer for stores.qa.stores.com.
If that's not an option, you could use an iRule to set the hostname based on the selected server. nmenant posted an example in this post (
Click here).
Aaron