Add Custom Domain: DNS record for the subdomain

Your admin is right. You need to use a new subdomain for the GTM Server Side container.
For example, if you have unbounce.example.com then you should set up a subdomain as gtm.unbounce.example.com or gtm.example.com depends on your needs.

This topic relates to this one How do I link the GTM-server IP to subdomain on shopify?