This article goes into detail about private link DNS configuration: https://learn.microsoft.com/en-us/azure/private-link/private-endpoint-dns
The FQDN you see in the portal is linking to the public DNS zone which you have blocked via disallowing public network access. Therefore the 3 options to use the FQDN are:
- Host file configuration (not recommended)
- Private DNS zone
- DNS forwarder
Outside of this you'll likely be working in unsupported territory.