The Domain Controller doesn’t host DHCP. The DHCP is managed by a Cisco router. Is there a way to make DHCP delegations visible to the Domain Controller.
The goal is to use a computer’s name to access it through DNS without having to manually set the DNS record and give out static IPs.
You can have the client machine register its DNS record(s) in active directory DNS, rather than the DHCP server doing it.
I can’t remember all of the steps off the top of my head, but google should help here. Essentially via group policy you can configure the clients to register their names in DNS. You should be able to find the relevant options under the network configuration settings. Those same settings can be tweaked via group policy.