Adding pihole DNS to router changes router admin and DHCP pool

The preferred method depends on if you can alter LAN DHCP DNS settings on the router.
If you can, first check if a client can query Pi-hole and gets a reply when you run below on a Windows, MacOS or Linux client (10.0.0.5 below is my Pi-hole IP):

C:\>nslookup pi.hole 10.0.0.5
Server:  pi.hole
Address:  10.0.0.5

Name:    pi.hole.home.dehakkelaar.nl
Addresses:  fe80::XXX:XXX:XXX:XXX
          10.0.0.5

Next check if it blocks a known ad domain (reply should be 0.0.0.0):

C:\>nslookup flurry.com 10.0.0.5
Server:  pi.hole
Address:  10.0.0.5

Name:    flurry.com
Addresses:  ::
          0.0.0.0

Next check if a public domain resolves:

C:\>nslookup pi-hole.net 10.0.0.5
Server:  pi.hole
Address:  10.0.0.5

Non-authoritative answer:
Name:    pi-hole.net
Address:  3.18.136.52

Only when those queries reply positive, you can proceed by changing only the router LAN section DHCP DNS setting.
No other router settings needs changing!
Pi-hole should be the sole DNS server for your clients and no others like a fallback DNS:

Next have the PC that you use for testing renews its DHCP lease.
They usually do when you shortly disconnect them from network.
And run below nslookup again but this time without the Pi-hole IP as an extra argument:

C:\>nslookup pi.hole
Server:  pi.hole
Address:  10.0.0.5

Name:    pi.hole.home.dehakkelaar.nl
Addresses:  fe80::XXX:XXX:XXX:XXX
          10.0.0.5

If that all works, you'd have to let the rest of your clients renew their DHCP leases.
You can do that all at once by power cycling your router, Wifi AP's and or network switches.
As soon as the clients detect the restored network connection, they will request a new DHCP lease.

Above is only for the IPv4 part!
IPv6 is another monster :wink:

Docs: