It would be Nice if I can configure for a client an other DNS Server via DHCP then the one for the specific network. 
 

On a linux server you can achieve this via the ISC DHCP server, there it would look llike in the config:

host some-client {
  # The client's MAC address
  hardware ethernet 00:11:22:33:44:55;
  
  # Optional: Assign a fixed IP to this client
  fixed-address 192.168.1.50;
  
  # Client-specific DNS options
  option domain-name-servers 1.1.1.1, 1.0.0.1;
}