Client 192.168.1.1 has been rate-limited (current config allows up to 1000 queries in 60 seconds)

Help
8

Above sounds like the Pi is not configured with static IP details yet.
So after you disabled the router DHCP service for Pi-hole to take over, the Pi still has an active DHCP lease from the router DHCP service for the duration of the DHCP lease period.
Depending router, this is usually 12 or 24 hours and can be checked with below:

$ sudo pihole-FTL dhcp-discover
Scanning all your interfaces for DHCP servers
[..]
   lease-time: 86400 ( 1d )

So after the lease-time expires, the Pi looses its IP details as it cant supply itself with IP details from its own DHCP service.

You can check with below one:

nmcli -t -f name con show --active | xargs -d '\n' -n 1 nmcli -p -f ipv4.method con show

If the Pi is still configured to acquire IP details via DHCP (the default) it will look like below (with "Wired connection 1" being my profile name for my network connection):

$ nmcli -t -f name con show --active | xargs  -d '\n' -n 1 nmcli -p -f ipv4.method con show
===============================================================================
                Connection profile details (Wired connection 1)
===============================================================================
ipv4.method:                            auto
-------------------------------------------------------------------------------
===============================================================================
                        Connection profile details (lo)
===============================================================================
ipv4.method:                            manual
-------------------------------------------------------------------------------

If the Pi is properly configured with a static IP instead, it will look like below with "manual" instead of "auto":

$ nmcli -t -f name con show --active | xargs  -d '\n' -n 1 nmcli -p -f ipv4.method con show
===============================================================================
                Connection profile details (Wired connection 1)
===============================================================================
ipv4.method:                            manual
-------------------------------------------------------------------------------
===============================================================================
                        Connection profile details (lo)
===============================================================================
ipv4.method:                            manual
-------------------------------------------------------------------------------

Dont mind the "lo" profile!
Thats for your loopback interface: