You can use your Pi-Hole as the DHCP server:
Your debug log shows no IP addresses assigned to the Pi. Pi-Hole is listening on 192.168.0.22, but that is not the IP of the Pi. Check in your router settings and verify that the Pi has a reserved IP of 22.
*** [ DIAGNOSING ]: Networking
[✗] No IPv4 address(es) found on the eth0 interface.
[✗] No IPv6 address(es) found on the eth0 interface.
[i] Default IPv4 gateway: 192.168.0.1
* Pinging 192.168.0.1...
[✗] Gateway did not respond. (https://discourse.pi-hole.net/t/why-is-a-default-gateway-important-for-pi-hole/3546)