DNS stops resolving very quickly after enabling pi-hole

If those are the only DNS settings in the router, they appear to be for the WAN side.
Which explains why your router WAN IP (your public Internet IP) appears in the Pi-hole logs.
Plus it explains the warning you received.
What is preferred and described in the docs is to have the DHCP service for your LAN (currently active on your router) to advertise the Pi-hole IP to its clients for DNS resolution.
Advantages are:
one less hop in the DNS path;
you see real stats from your clients on the webGUI instead of only your router;
and particular Pi-hole features become available like for example defining client groups.

Check if you can disable the DHCP service on the router?
As a replacement, you can let Pi-hole take over the DHCP service part for your network:

Best when switching is to enable the DHCP service on Pi-hole first before disabling the one on the router!
This because some routers drop the LAN connection temporarily causing your clients to try and renew their DHCP lease while the DHCP service might not be up and running yet.

Have you made sure that your Pi-hole instance is not an open resolver?
You can check by running below one on the Pi to figure out your public IP:

curl ifconfig.me; echo

And enter the public IP below to check:

https://www.openresolver.com/