It's actually crucial that the FQDN for the core is not resolvable out in Internet-land as this is one of the first checks LANDesk does to determine if the core is reachable or not. So your core and gateway should have their own unique names.
It is possible that having an alias for your core will effect connectivity through the gateway. The core name is baked into the cert that is passed and if these are mismatched it can cause the connection to fail. So, the simplest approach is to use one name for your core.
And don't change the ip address or the fqdn on the core after it's built. Otherwise you're going to be rebuilding it.