I would imagine to properly troubleshoot this, we would need an HTTP trace from Fiddler
. On the local LAN, Windows will transparently pass credentials through the WinInet layer. If the UAAS URL is not in the local intranet zone, for instance if your Windows domain is mycompany.local and your UAAS is mailarchive.mycompany.com, you will have to add the UAAS URL to each computer's Local Intranet zone explicitly or you will be prompted for credentials.
The only reason I can think of for multiple prompts is that you have a web proxy between you and the UAAS, and that proxy does not have power to delegate credentials to the UAAS.
This is all entirely dependent on your network topology, which we know nothing about. I would recommend trying to trace the network activity using Fiddler, and from there you can configure any server inbetween your user and the UAAS so that it can pass credentials.