This shouldn’t be a problem. Just set up the Pi to use a static IP, and configure your router to give out the Pi’s IP address as the DNS server to the DHCP clients.
To configure a static IP, do the following:
connmanctl services
This will return the connman service name of your ethernet adapter. Let’s call it EthernetName.
Then,
connmanctl config EthernetName ipv4 manual address netmask gateway