Solution:
It turned out Router 1 has a LAN/WAN Port. As soon as I plugged Pi-Hole into that Port, everything started working.
Final working wiring setup:
- There's a wired ethernet connection between Router 1 LAN/WAN Port and Pi-Hole (DHCP).
- There's a wired ethernet connection between Router 1 LAN 1 and Router 2 LAN 1.
Thanks again.