Integrated authentication will work as long as integrated authentication works.
IE is probably prompting because:
- the target site name contains dots, and
- the target site isn't in the Local Intranet Zone
- and possibly that the target site is being proxied (due to #1 or resulting in #2 or because of WPAD etc)
Then there's the Kerberos issue - i.e. if you require Kerberos, you'll need SPNs configured for that domain name.
But fundamentally, IE should provide transparent logon with the logged-on user's credentials to any site it considers part of the Local Intranet zone, with default settings applied to it.
Note Trusted Sites sounds better, but it doesn't permit automatic logon by default. From memory.