Having proven the problem is name resolution, I’d still try to avoid manual HOSTS file updates. That would be an ongoing maintenance problem in the future.
If this is a brand-new deployment, I’d start over, changing the license to reference a fully-qualified host name for the root server, that can be resolved in DNS.
Another option would be to use the BESAdmin tool to define a “Last Fallback Relay”, which should be a fully-qualified DNS name that points to either a Relay that all clients can resolve and reach, or the fully-qualified name to the root server itself.
You could also add DNS Aliases to whatever domains your clients have in their DNS Suffix Search Order (see /etc/resolve.conf) so the short name resolves to your clients. That may mean multiple aliases if your clients have different DNS search orders.
Or you can define a manual relay or manual relay list in the advanced options of the Client Deploy Tool,so the clients register with a specific (resolvable) relay when they are installed.